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There are two kinds of models in mathematical ecology, broadly speaking. There are, on the one hand, models of strategic type, which are based on empirical formulas and use computer simulation techniques. These are popular among ecologists because they fit the data extremely well and are highly predictive in particular cases, say a wheat field in Saskatchewan or a sheep herd in New Zealand. But, in fact, they tell next to nothing about the underlying ecology. On the other hand, there are dynamical models, which often involve ordinary differential equations, but may use stochastic differential equations, difference equations, integral equations, or diffusion reaction equations. These models encode postulates about ecological mechanisms into the equations. As a rule, these do not predict as well as strategic models do, because of the constraints imposed by these postulates. But it is through the use of dynamical models that tentative explanations can be found and eventual consensus reached, so that more general, improved, strategic models can be designed for ecosystem management. Below, for the sake of brevity only ordinary differential equation models are considered.
 
There are two kinds of models in mathematical ecology, broadly speaking. There are, on the one hand, models of strategic type, which are based on empirical formulas and use computer simulation techniques. These are popular among ecologists because they fit the data extremely well and are highly predictive in particular cases, say a wheat field in Saskatchewan or a sheep herd in New Zealand. But, in fact, they tell next to nothing about the underlying ecology. On the other hand, there are dynamical models, which often involve ordinary differential equations, but may use stochastic differential equations, difference equations, integral equations, or diffusion reaction equations. These models encode postulates about ecological mechanisms into the equations. As a rule, these do not predict as well as strategic models do, because of the constraints imposed by these postulates. But it is through the use of dynamical models that tentative explanations can be found and eventual consensus reached, so that more general, improved, strategic models can be designed for ecosystem management. Below, for the sake of brevity only ordinary differential equation models are considered.
  
 
==Growth of a single population.==
 
==Growth of a single population.==
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m1201301.png" /> denote the total number, or density, of a population <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m1201302.png" /> at a fixed location and time. Assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m1201303.png" /> is continuous in the time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m1201304.png" />. The Hutchinson postulates [[#References|[a16]]] are:
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Let $N ( t )$ denote the total number, or density, of a population $\Sigma$ at a fixed location and time. Assume that $N ( t )$ is continuous in the time $t$. The Hutchinson postulates [[#References|[a16]]] are:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m1201305.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m1201306.png" /> sufficiently differentiable;
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1) $d N / d t = f ( N )$, $f$ sufficiently differentiable;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m1201307.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m1201308.png" />;
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2) $N \equiv 0$ implies $d N / d t \equiv 0$;
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m1201309.png" /> is bounded between zero and a fixed positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013010.png" />, for all time.
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3) $N ( t )$ is bounded between zero and a fixed positive constant $C$, for all time.
  
Given the Hutchinson postulates for a population <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013011.png" />, it follows that the ordinary differential equation
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Given the Hutchinson postulates for a population $\Sigma$, it follows that the ordinary differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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\begin{equation} \tag{a1} \frac { d N } { d t } = \lambda N \left( 1 - \frac { N } { K } \right) , \end{equation}
  
 
for which
 
for which
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
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\begin{equation} \tag{a2} N ( t ) = \frac { K } { 1 + b e ^ { - \lambda t } } \end{equation}
  
 
is the general solution, is the simplest growth law. It is called the logistic equation.
 
is the general solution, is the simplest growth law. It is called the logistic equation.
  
The parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013014.png" />, called the carrying capacity for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013015.png" />, obviously satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013016.png" />. The parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013017.png" /> is called the intrinsic growth rate. Of the four types of shapes specified for (a1) by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013021.png" />, only the last is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013022.png" />-shaped (i.e. its graph has an inflection point).
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The parameter $K$, called the carrying capacity for $\Sigma$, obviously satisfies $0 &lt; K \leq C$. The parameter $\lambda &gt; 0$ is called the intrinsic growth rate. Of the four types of shapes specified for (a1) by $b &lt; 0$, $b = 0$, $0 &lt; b \leq 1$, $b &gt; 1$, only the last is $S$-shaped (i.e. its graph has an inflection point).
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013023.png" /> satisfies only (a1) and (a2); then, denoting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013025.png" /> (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013026.png" /> is a steady-state), Taylor expansion around <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013027.png" /> gives
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Suppose that $\Sigma$ satisfies only (a1) and (a2); then, denoting $n ( t ) = N ( t ) - N_ {*}$, where $f ( N_{ *} ) = 0$ (i.e. $N _{*}$ is a steady-state), Taylor expansion around $N _{*}$ gives
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013028.png" /></td> </tr></table>
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\begin{equation*} \frac { d N } { d t } = \frac { d n } { d t } = f ( N ) = \end{equation*}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013029.png" /></td> </tr></table>
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\begin{equation*} = f ( N_{ * } ) + f ^ { \prime } ( N_{ * } ) n + \frac { f ^ { \prime \prime } ( N_{ * } ) } { 2 } n ^ { 2 } + \ldots, \end{equation*}
  
where the prime denotes differentiation with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013030.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013031.png" /> small in absolute value, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013032.png" /> is well approximated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013033.png" />. Therefore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013034.png" /> increases with time if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013035.png" />, and decreases if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013036.png" />. In the former case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013037.png" /> is an unstable steady-state while, in the latter case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013038.png" /> is a stable steady-state.
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where the prime denotes differentiation with respect to $N$. For $n ( t )$ small in absolute value, $d n / d t$ is well approximated by $f ^ { \prime } ( N_{*} ) n$. Therefore, $n ( t )$ increases with time if $f ^ { \prime } ( N _{*} ) &gt; 0$, and decreases if $f ^ { \prime } ( N_{*} ) &lt; 0$. In the former case, $N _{*}$ is an unstable steady-state while, in the latter case, $N _{*}$ is a stable steady-state.
  
For the logistic special case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013039.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013040.png" /> are the only possible steady-states, the former being unstable and the latter stable.
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For the logistic special case, $N_{*} = 0$ or $N_* = K$ are the only possible steady-states, the former being unstable and the latter stable.
  
 
The logistic differential equation (a1) is the simplest description of a population with limited resources, the limitation being provided by the negative coefficient of the quadratic term. The equation first arose in the work of P. Verhulst (1838) and later in the demographic research of R. Pearl and L. Reed in the 1920s. It was subsequently used to provide a dynamic model of malaria in humans by Sir Ronald Ross, but has perhaps a more basic role in ecology than in epidemiology.
 
The logistic differential equation (a1) is the simplest description of a population with limited resources, the limitation being provided by the negative coefficient of the quadratic term. The equation first arose in the work of P. Verhulst (1838) and later in the demographic research of R. Pearl and L. Reed in the 1920s. It was subsequently used to provide a dynamic model of malaria in humans by Sir Ronald Ross, but has perhaps a more basic role in ecology than in epidemiology.
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Several species living in the same locality must forage for food and seek nesting sites in a field or stream, etc. These populations may or may not affect one another.
 
Several species living in the same locality must forage for food and seek nesting sites in a field or stream, etc. These populations may or may not affect one another.
  
Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013041.png" /> species comprise a community <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013042.png" /> in which there are no inter-specific interactions. This ecosystem can be modeled by
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Suppose that $n$ species comprise a community $\Sigma$ in which there are no inter-specific interactions. This ecosystem can be modeled by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013043.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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\begin{equation} \tag{a3} \frac { d N ^ { i } } { d t } = \lambda _ { ( i ) } N ^ { i } \left( 1 - \frac { N ^ { i } } { K _ { ( i ) } } \right) , \quad i = 1 , \ldots , n, \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013044.png" /> denotes the total number or density of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013045.png" />th species in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013046.png" />. This system has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013047.png" /> steady-states, but only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013048.png" /> is stable. The equations (a3) describe non-competition.
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where $N ^ { i }$ denotes the total number or density of the $i$th species in $\Sigma$. This system has $2 ^ { n }$ steady-states, but only $( K _ { ( 1 ) } , \dots , K _ { ( n ) } )$ is stable. The equations (a3) describe non-competition.
  
Now suppose there is competition for food items, etc. How does one describe this? G.F. Gause and A.A. Witt answered this for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013049.png" />-species community (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013050.png" />) with [[#References|[a11]]]
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Now suppose there is competition for food items, etc. How does one describe this? G.F. Gause and A.A. Witt answered this for a $2$-species community ($n = 2$) with [[#References|[a11]]]
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
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\begin{equation} \tag{a4} \left\{ \begin{array}{l}{ \frac { d N ^ { 1 } } { d t } = \lambda _ { ( 1 ) } N ^ { 1 } \left( 1 - \frac { N ^ { 1 } } { K _ { ( 1 ) } } - \delta _ { ( 1 ) } \frac { N ^ { 2 } } { K _ { ( 1 ) } } \right), }\\{ \frac { d N ^ { 2 } } { d t } = \lambda _ { ( 2 ) } N ^ { 2 } \left( 1 - \frac { N ^ { 2 } } { K _ { ( 2 ) } } - \delta _ { ( 2 ) } \frac { N ^ { 1 } } { K _ { ( 2 ) } } \right). }\end{array} \right. \end{equation}
  
Here, all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013054.png" /> are positive. This system has exactly one positive equilibrium <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013055.png" />, given by
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Here, all $\lambda$, $K$ and $\delta$ are positive. This system has exactly one positive equilibrium $( N _ { * } ^ { 1 } , N _ { * } ^ { 2 } )$, given by
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013056.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
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\begin{equation} \tag{a5} \left\{ \begin{array}{l}{ N _ { * } ^ { 1 } = \frac { K _ { ( 1 ) } - \delta _ { ( 1 ) } K _ { ( 2 ) } } { 1 - \delta _ { ( 1 ) } \delta _ { ( 2 ) } }, }\\{ N _ { * } ^ { 2 } = \frac { K _ { ( 2 ) } - \delta _ { ( 2 ) } K _ { ( 1 ) } } { 1 - \delta _ { ( 1 ) } \delta _ { ( 2 ) } }. }\end{array} \right. \end{equation}
  
If both numerators and denominators are positive, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013057.png" /> in (a5) is stable. If they are both negative, (a5) is unstable. This is easily proved by using the stability Ansatz: the eigenvalues of the Jacobian of the right-hand side of a system
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If both numerators and denominators are positive, then $( N _ { * } ^ { 1 } , N _ { * } ^ { 2 } )$ in (a5) is stable. If they are both negative, (a5) is unstable. This is easily proved by using the stability Ansatz: the eigenvalues of the Jacobian of the right-hand side of a system
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013058.png" /></td> </tr></table>
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\begin{equation*} \frac { d N ^ { i } } { d t } = f ^ { i } ( N ^ { 1 } , \ldots , N ^ { n } ) , \quad i = 1 , \dots , n, \end{equation*}
  
evaluated at a steady-state <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013059.png" />, must have negative real part for stability. If any of these is positive, an unstable case results.
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evaluated at a steady-state $( N _ { * } ^ { 1 } , \ldots , N _ { * } ^ { n } )$, must have negative real part for stability. If any of these is positive, an unstable case results.
  
 
In the question of survival for the two populations in Gause–Witt competition (a4), (a5), there are four cases to consider:
 
In the question of survival for the two populations in Gause–Witt competition (a4), (a5), there are four cases to consider:
  
A) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013060.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013061.png" />, then (a5) is unstable, with survival depending on the initial proportions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013063.png" />.
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A) If $\delta _{( 1 )}  &gt; K _ { ( 1 ) } / K _ { ( 2 ) }$ and $\delta_{( 2 )} &gt; K _ { ( 2 ) } / K _ { ( 1 ) }$, then (a5) is unstable, with survival depending on the initial proportions of $N ^ { 1 }$ and $N ^ { 2 }$.
  
B) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013065.png" />, then (a5) is unstable, and the first species will be eliminated.
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B) If $\delta _{( 1 )}  &gt; K _ { ( 1 ) } / K _ { ( 2 ) }$ and $\delta _ { ( 2 ) } &lt; K _ { ( 2 ) } / K _ { ( 1 ) }$, then (a5) is unstable, and the first species will be eliminated.
  
C) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013067.png" />, then (a5) is unstable, and the second species will be eliminated.
+
C) If $\delta _ { ( 1 ) } &lt; K _ { ( 1 ) } / K _ { ( 2 ) }$ and $\delta_{( 2 )} &gt; K _ { ( 2 ) } / K _ { ( 1 ) }$, then (a5) is unstable, and the second species will be eliminated.
  
D) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013068.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013069.png" />, then (a5) is stable.
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D) If $\delta _ { ( 1 ) } &lt; K _ { ( 1 ) } / K _ { ( 2 ) }$ and $\delta _ { ( 2 ) } &lt; K _ { ( 2 ) } / K _ { ( 1 ) }$, then (a5) is stable.
  
 
Therefore, only in case D), called incomplete competition, can both species coexist. This case translates as some geometrical separation of the two species, where the more vulnerable one has a refuge it can retreat to, or some resource available that the otherwise better adapted competitor cannot use [[#References|[a16]]].
 
Therefore, only in case D), called incomplete competition, can both species coexist. This case translates as some geometrical separation of the two species, where the more vulnerable one has a refuge it can retreat to, or some resource available that the otherwise better adapted competitor cannot use [[#References|[a16]]].
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==Three-species interactions: a general model applicable to several different ecosystems.==
 
==Three-species interactions: a general model applicable to several different ecosystems.==
One of the great benefits of dynamical models is their tendency to be applicable in more than one ecological situation. This is partly because they are framed in precise mathematical terms encoding a list of specific postulates and assumptions, but also because in their qualitative behaviour lies the essence of their application. An illustration of this is the example given below, of a model known to encompass three different ecosystems. The model exhibits switching between multiple steady-states and stable periodic solutions (i.e. stable limit cycles) induced by predation of one species on another. In its full generality, the system (a9) models predation on a herbivore which in turn feeds on a plant species. The limit cycle behaviour described is not induced by time-lags, as in the classical Lotka–Volterra predator-prey model (with predator devastating the prey population to the extent that there is not enough prey for the much larger predator population, which then crashes, resulting in the prey population coming back full circle). Rather, the mechanism is aggregation, caused by spawning or feeding behaviour of the predator, conditioned by environmental constraints in some cases (e.g. cyclones, drought, nutrient enrichment, etc.). Furthermore, one must always prove that a periodic solution, topologically a circle, is stable, in the sense that there is a solid torus <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013070.png" /> in phase-space whose centre is the cycle and having the property that any solution with initial conditions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013071.png" /> will converge onto that cycle as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013072.png" />. The methods of [[Hopf bifurcation|Hopf bifurcation]] provide the necessary tools for this analysis [[#References|[a15]]].
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One of the great benefits of dynamical models is their tendency to be applicable in more than one ecological situation. This is partly because they are framed in precise mathematical terms encoding a list of specific postulates and assumptions, but also because in their qualitative behaviour lies the essence of their application. An illustration of this is the example given below, of a model known to encompass three different ecosystems. The model exhibits switching between multiple steady-states and stable periodic solutions (i.e. stable limit cycles) induced by predation of one species on another. In its full generality, the system (a9) models predation on a herbivore which in turn feeds on a plant species. The limit cycle behaviour described is not induced by time-lags, as in the classical Lotka–Volterra predator-prey model (with predator devastating the prey population to the extent that there is not enough prey for the much larger predator population, which then crashes, resulting in the prey population coming back full circle). Rather, the mechanism is aggregation, caused by spawning or feeding behaviour of the predator, conditioned by environmental constraints in some cases (e.g. cyclones, drought, nutrient enrichment, etc.). Furthermore, one must always prove that a periodic solution, topologically a circle, is stable, in the sense that there is a solid torus $T$ in phase-space whose centre is the cycle and having the property that any solution with initial conditions in $T$ will converge onto that cycle as $t \rightarrow + \infty$. The methods of [[Hopf bifurcation|Hopf bifurcation]] provide the necessary tools for this analysis [[#References|[a15]]].
  
The logistic growth equation with exponential parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013073.png" />,
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The logistic growth equation with exponential parameter $a &gt; 0$,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013074.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
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\begin{equation} \tag{a6} \frac { d N } { d t } = \lambda N \left( 1 - \left( \frac { N } { K } \right) ^ {a } \right), \end{equation}
  
was introduced to explain certain data on Drosophila in [[#References|[a13]]]. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013075.png" /> indicates greater self-inhibition while the converse is true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013076.png" />. Similarly, the dynamical model
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was introduced to explain certain data on Drosophila in [[#References|[a13]]]. The case $a &gt; 1$ indicates greater self-inhibition while the converse is true for $a &lt; 1$. Similarly, the dynamical model
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013077.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>
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\begin{equation} \tag{a7} \frac { d F } { d t } = - \varepsilon F ( 1 - \gamma F ^ { p } ), \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013078.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013080.png" />, was introduced to explain crown-of-thorns starfish (Acanthaster planci) aggregation on coral reefs [[#References|[a1]]], [[#References|[a21]]]. The term <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013081.png" /> is called the cooperative term. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013082.png" />, then the variable coefficient of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013083.png" /> in (a7) is relatively large for small values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013084.png" />. This results in increased cooperation, and the reverse is true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013085.png" />. The parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013086.png" /> is the coefficient of aggregation. It also serves as Hopf bifurcation parameter in (a8) and (a9), where Hopf's method can be used to prove the existence of small amplitude-stable periodic solutions (i.e. stable limit cycles), [[#References|[a15]]]. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013087.png" /> is fixed in a model, unlike <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013088.png" />, which is a free parameter. Rather, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013089.png" /> is an indicator of fecundity or genetically determined potential for reproduction. The role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013090.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013091.png" /> in (a8) and (a9) is investigated below.
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where $\varepsilon &gt; 0$, $\gamma &gt; 0$ and $p \in ( 1 / 2,3 / 2 )$, was introduced to explain crown-of-thorns starfish (Acanthaster planci) aggregation on coral reefs [[#References|[a1]]], [[#References|[a21]]]. The term $\gamma F ^ { p }$ is called the cooperative term. If $p &lt; 1$, then the variable coefficient of $F ^ { 2 }$ in (a7) is relatively large for small values of $F$. This results in increased cooperation, and the reverse is true for $p &gt; 1$. The parameter $\gamma$ is the coefficient of aggregation. It also serves as Hopf bifurcation parameter in (a8) and (a9), where Hopf's method can be used to prove the existence of small amplitude-stable periodic solutions (i.e. stable limit cycles), [[#References|[a15]]]. Note that $p$ is fixed in a model, unlike $\gamma$, which is a free parameter. Rather, $p$ is an indicator of fecundity or genetically determined potential for reproduction. The role of $p$ and $\gamma$ in (a8) and (a9) is investigated below.
  
 
Consider the ordinary differential equations
 
Consider the ordinary differential equations
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013092.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a8)</td></tr></table>
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\begin{equation} \tag{a8} \left. \begin{cases}  { \frac { d N } { d t } = N ( - 2 \alpha N - \delta F + \lambda ), } \\ { \frac { d F } { d t } = F ( 2 \beta N + \gamma F ^ { p } - \varepsilon ), } \end{cases} \right. \end{equation}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013093.png" /> is taken slightly less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013094.png" />. The constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013098.png" /> can be given a precisely defined chemical interpretation based on the concepts of the Volterra production variable and on the Rhoades allometric plant response mechanism [[#References|[a19]]], [[#References|[a21]]]; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m12013099.png" /> denotes the density of plant modular units (e.g. leaves); [[#References|[a14]]]. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130100.png" /> is the density of the herbivore population in the same locality. The system (a8) is a model in the theory of optimal defense of plants against herbivores, [[#References|[a19]]].
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where $p$ is taken slightly less than $1$. The constants $\alpha$, $\delta$, $\beta$, $\gamma$ can be given a precisely defined chemical interpretation based on the concepts of the Volterra production variable and on the Rhoades allometric plant response mechanism [[#References|[a19]]], [[#References|[a21]]]; $N$ denotes the density of plant modular units (e.g. leaves); [[#References|[a14]]]. $F$ is the density of the herbivore population in the same locality. The system (a8) is a model in the theory of optimal defense of plants against herbivores, [[#References|[a19]]].
  
Use of Hopf bifurcation theory and the Hassard code BIFOR2 show the existence of a stable periodic solution (i.e. limit cycle) of small amplitude [[#References|[a1]]], [[#References|[a15]]]. One may also show that the amplitude can be large [[#References|[a3]]]. It is also possible to show that the period of the cycle is longer for plants which use the metabolically expensive chemical defense (e.g. oaks), as opposed to plants (e.g. herbaceous) which do not. This explains both the 9–10 year cycle of the oak caterpillar and the 3–4 cycle of voles and lemmings which eat herbaceous plants. The model requires <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130101.png" /> so that the herbivore must not only have highly aggregative behaviour, but must be highly fecund.
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Use of Hopf bifurcation theory and the Hassard code BIFOR2 show the existence of a stable periodic solution (i.e. limit cycle) of small amplitude [[#References|[a1]]], [[#References|[a15]]]. One may also show that the amplitude can be large [[#References|[a3]]]. It is also possible to show that the period of the cycle is longer for plants which use the metabolically expensive chemical defense (e.g. oaks), as opposed to plants (e.g. herbaceous) which do not. This explains both the 9–10 year cycle of the oak caterpillar and the 3–4 cycle of voles and lemmings which eat herbaceous plants. The model requires $p \ll 1$ so that the herbivore must not only have highly aggregative behaviour, but must be highly fecund.
  
An interesting application of the chemically mediated plant/herbivore system (a8) is to the lynx-snowshoe hare (Lepus americanus) cycle in the Arctic (cf. also [[Canadian lynx data|Canadian lynx data]]; [[Canadian lynx series|Canadian lynx series]]). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130102.png" /> denotes the modular unit density for the plant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130103.png" /> the hare density. The large reproductive potential of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130104.png" />-population is interpreted as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130105.png" />. It is known that the plants which hares eat are chemically defended and that this has a strong negative effect on the hare population. It was discovered in the field that the hare population cycles both with and without the presence of lynx [[#References|[a8]]], [[#References|[a17]]], [[#References|[a6]]]! The three species extension of (a8), which incorporates the lynx, is
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An interesting application of the chemically mediated plant/herbivore system (a8) is to the lynx-snowshoe hare (Lepus americanus) cycle in the Arctic (cf. also [[Canadian lynx data|Canadian lynx data]]; [[Canadian lynx series|Canadian lynx series]]). $N$ denotes the modular unit density for the plant and $F$ the hare density. The large reproductive potential of the $F$-population is interpreted as $p \ll 1$. It is known that the plants which hares eat are chemically defended and that this has a strong negative effect on the hare population. It was discovered in the field that the hare population cycles both with and without the presence of lynx [[#References|[a8]]], [[#References|[a17]]], [[#References|[a6]]]! The three species extension of (a8), which incorporates the lynx, is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130106.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a9)</td></tr></table>
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\begin{equation} \tag{a9}  \begin{cases} { l } { \frac { d N } { d t } = N ( - 2 \alpha N - \delta F + \lambda ) }, \\ { \frac { d F } { d t } = F ( 2 \beta N + \gamma F ^ { p } - \varepsilon - \mu _ { 1 } L ) }, \\ { \frac { d L } { d t } = \mu _ { 2 } L F - \nu L }, \end{cases}  \end{equation}
  
where all constants are non-negative. For convenience one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130107.png" />, but this has no biological significance. However, if one also sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130108.png" /> and rewrites the third equation as
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where all constants are non-negative. For convenience one sets $\alpha = \beta$, but this has no biological significance. However, if one also sets $\mu _ { 2 } = \gamma$ and rewrites the third equation as
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a10)</td></tr></table>
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\begin{equation} \tag{a10} \frac { d L } { d t } = \gamma L ( F - \xi ) , \quad \xi = \frac { \nu } { \gamma }, \end{equation}
  
the assumption <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130110.png" /> implies that the predator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130111.png" /> is getting more food value out of its kill, all other things being equal, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130112.png" /> increases. This model shows that the high reproductivity of the Arctic hare population (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130113.png" />) drives a stable periodic cycle whose period increases with increasing amounts of defensive compounds in the plant tissues. Also, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130114.png" />-population will cycle without the lynx and so the lynx-hare cycle is driven by the hare's food quality, with the lynx population going along piggy-back style. This model is an improvement over the time-lag model, [[#References|[a20]]], [[#References|[a12]]].
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the assumption $\mu _ { 2 } = \gamma$ implies that the predator $L$ is getting more food value out of its kill, all other things being equal, as $\gamma$ increases. This model shows that the high reproductivity of the Arctic hare population ($p \ll 1$) drives a stable periodic cycle whose period increases with increasing amounts of defensive compounds in the plant tissues. Also, the $F$-population will cycle without the lynx and so the lynx-hare cycle is driven by the hare's food quality, with the lynx population going along piggy-back style. This model is an improvement over the time-lag model, [[#References|[a20]]], [[#References|[a12]]].
  
A model of Acanthaster planci predation on corals of the Great Barrier Reef is provided by (a8), but without the chemical interpretation for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130115.png" />-population, which in this case is coral. The starfish population is highly fecund and aggregates, causing outbreaks with a 12–15 year period [[#References|[a4]]]. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130116.png" /> will generate a bifurcation from the positive equilibrium of (a8) to a stable periodic cycle triggered by increasing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130117.png" /> beyond a certain critical Hopf value determined by the coefficients in (a8). The extended system (a9) can be used to discuss the claim of marine biologist R. Endean that the giant conch, C. tritonis, which preys on adult Acanthaster plani, may be a keystone predator on the Great Barrier Reef [[#References|[a9]]]. Such a conception excludes any limit cycle behaviour, a priori, and is essentially a steady-state theory. Assuming that C. tritonis gains when starfish aggregate (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130118.png" /> increases) and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130119.png" />, the model (a9) predicts Hopf bifurcation from a steady-state to a stable limit cycle of moderate amplitude. Consequently, C. tritonis must also cycle synchronously (i.e. piggy-back). However, there is no evidence for regular conch fluctuations in this case. Yet, if A. planci were neither highly fecund nor aggregative, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130120.png" /> would have to be used in (a9) and the result would be a steady-state (perhaps several). That is, giant triton would be a keystone predator, similar to the role of sea otters in the Western Canadian sea urchin-kelp system discussed below.
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A model of Acanthaster planci predation on corals of the Great Barrier Reef is provided by (a8), but without the chemical interpretation for the $N$-population, which in this case is coral. The starfish population is highly fecund and aggregates, causing outbreaks with a 12–15 year period [[#References|[a4]]]. Thus, $p \ll 1$ will generate a bifurcation from the positive equilibrium of (a8) to a stable periodic cycle triggered by increasing $\gamma$ beyond a certain critical Hopf value determined by the coefficients in (a8). The extended system (a9) can be used to discuss the claim of marine biologist R. Endean that the giant conch, C. tritonis, which preys on adult Acanthaster plani, may be a keystone predator on the Great Barrier Reef [[#References|[a9]]]. Such a conception excludes any limit cycle behaviour, a priori, and is essentially a steady-state theory. Assuming that C. tritonis gains when starfish aggregate (i.e. $\gamma$ increases) and that $p \ll 1$, the model (a9) predicts Hopf bifurcation from a steady-state to a stable limit cycle of moderate amplitude. Consequently, C. tritonis must also cycle synchronously (i.e. piggy-back). However, there is no evidence for regular conch fluctuations in this case. Yet, if A. planci were neither highly fecund nor aggregative, then $p \geq 1$ would have to be used in (a9) and the result would be a steady-state (perhaps several). That is, giant triton would be a keystone predator, similar to the role of sea otters in the Western Canadian sea urchin-kelp system discussed below.
  
On the west coast of North America, red sea urchins (S. franciscanus) feed on kelps in large aggregates and exist in at least two possible steady-states: at very low density within kelp beds (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130121.png" />) in the presence of sea otters; or at high density outside kelp beds (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130122.png" />) in the absence of sea otters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130123.png" /> [[#References|[a5]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130124.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130125.png" /> is relatively large, as is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130126.png" />. The system (a9) has a unique positive equilibrium for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130127.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130128.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130129.png" />. It is
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On the west coast of North America, red sea urchins (S. franciscanus) feed on kelps in large aggregates and exist in at least two possible steady-states: at very low density within kelp beds ($\delta \neq 0$) in the presence of sea otters; or at high density outside kelp beds ($\delta \approx 0$) in the absence of sea otters $( L _ { 0 } \approx 0 )$ [[#References|[a5]]]. If $L _ { 0 } \approx 0$, then $F _ { 0 }$ is relatively large, as is $N_ 0 $. The system (a9) has a unique positive equilibrium for $\lambda - \delta \xi &gt; 0$ and $2 \beta N _ { 0 } + \gamma \xi ^ { p } - \varepsilon &gt; 0$, $p &gt; 1$. It is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130130.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a11)</td></tr></table>
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\begin{equation} \tag{a11} N _ { 0 } = \frac { \lambda - \delta \xi } { 2 \alpha } , L _ { 0 } = \frac { 2 \beta N _ { 0 } + \gamma \xi ^ { p } - \varepsilon } { \mu _ { 1 } } , F _ { 0 } = \xi. \end{equation}
  
In the case where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130131.png" />, the system reduces to one with steady-state: (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130132.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130134.png" />), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m120/m120130/m120130135.png" />. It is known from field data that the steady-state can rapidly switch and depends only on the presence or absence of sea otters. The otter is a keystone predator causing rapid switching in the red sea urchin population.
+
In the case where $\delta \approx 0$, the system reduces to one with steady-state: ($N _ { 0 } = \lambda / ( 2 \alpha )$, $L _ { 0 } = 0$, $F _ { 0 } = \xi$), with $p &gt; 1$. It is known from field data that the steady-state can rapidly switch and depends only on the presence or absence of sea otters. The otter is a keystone predator causing rapid switching in the red sea urchin population.
  
 
The above model also applies to the lobster-sea urchin-kelp system of the Eastern Canadian coast. In this case the lobsters play the keystone predator role, [[#References|[a18]]].
 
The above model also applies to the lobster-sea urchin-kelp system of the Eastern Canadian coast. In this case the lobsters play the keystone predator role, [[#References|[a18]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> "Mathematical essays on growth and the emergence of form" P.L. Antonelli (ed.) , Univ. Alberta Press (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Antonelli,   R. Bradbury,   X. Lin,   "On Hutchinson's competition equations and their homogenization: A higher-order principle of competitive exclusion" ''Ecol. Modelling'' , '''60''' (1992) pp. 309–320</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P.L. Antonelli,   K.D. Fuller,   N.D. Kazarinoff,   "A study of large amplitude periodic solutions in a model of starfish predation on coral" ''IMA J. Math. Appl. in Medicine and Biol.'' , '''4''' (1987) pp. 207–214</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> "Acanthaster and the coral reef: A theoretical perspective" R. Bradbury (ed.) , ''Lecture Notes Biomath.'' , '''88''' , Springer (1990)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> P. Breen,   T.A. Caros,   J.B. Foster,   E.A. Stewart,   "Changes in subtidal community structure associated with British Columbia sea otter transplants" ''Marine Eco.'' , '''7''' (1982) pp. 13–20</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J.T. Bryant,   "The regulation of snowshoe hare feeding behaviour during winter by plant anti-herbivore chemistry" K. Myers (ed.) C.D. McInness (ed.) , ''Proc. World Lagomorph Conf.'' , Guelph Univ. Press (1979)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> N. Elredge,   "Time frames, the evolution of punctuated equilibria" , Princeton Univ. Press (1989)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> C.S. Elton,   "The ecology of invasion by animals and plants" , Methuen (1958)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> R. Endean,   "Acanthaster planci infestations of reefs of the Great Barrier Reef" , ''Proc. Third Internat. Coral Reef Symp.'' , '''1''' (1977) pp. 185–191</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> G.F. Gause,   "The struggle for existence" , Williams and Wilkins (1934)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> G.F. Gause,   A.A. Witt,   "Behaviour of mixed populations and the problem of natural selection" ''Amer. Nat.'' , '''69''' (1935) pp. 596–609</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> M.E. Gilpin,   "Do hares eat lynx?" ''Amer. Nat.'' , '''107''' (1973) pp. 727–730</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> M. Gilpin,   F.J. Ayala,   "Global models of growth and competition" ''Proc. Nat. Acad. Sci.'' , '''70''' (1973) pp. 3590–3593</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top"> J.L. Harper,   "The population biology of plants" , Acad. Press (1977)</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top"> B. Hassard,   N.D. Kazarinoff,   Y.-H. Wan,   "Theory and applications of Hopf bifurcations" , ''London Math. Soc. Lecture Notes'' , '''41''' , Cambridge Univ. Press (1981)</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top"> G.E. Hutchinson,   "An introduction to population biology" , Yale Univ. Press (1978)</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top"> L.B. Keith,   "Wildlife's ten-year cycle" , Univ. Wisconsin Press (1963)</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top"> K. Mann,   "Kelp, sea urchins and predators: a review of strong interactions in rocky subtidal systems of eastern Canada 1970–1980" ''Netherl. J. Sea Research'' , '''16''' (1982) pp. 414–423</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top"> D.F. Rhoades,   "Offensive-defensive interactions between herbivores and plants: their relevance in herbivore population dynamics and ecological theory" ''Amer. Nat.'' , '''125''' (1985) pp. 205–223</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top"> R.E. Ricklefs,   "Ecology" , Chiron Press (1979) (Edition: Second)</TD></TR><TR><TD valign="top">[a21]</TD> <TD valign="top"> P.L. Antonelli,   R. Bradbury,   "Volterra–Hamilton models in the ecology and evolution of colonial organisms" , ''Ser. Math. Biol. and Medicine'' , World Sci. (1996)</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top"> "Mathematical essays on growth and the emergence of form" P.L. Antonelli (ed.) , Univ. Alberta Press (1985) {{MR|0826076}} {{ZBL|0572.00026}} </td></tr><tr><td valign="top">[a2]</td> <td valign="top"> P. Antonelli, R. Bradbury, X. Lin, "On Hutchinson's competition equations and their homogenization: A higher-order principle of competitive exclusion" ''Ecol. Modelling'' , '''60''' (1992) pp. 309–320</td></tr><tr><td valign="top">[a3]</td> <td valign="top"> P.L. Antonelli, K.D. Fuller, N.D. Kazarinoff, "A study of large amplitude periodic solutions in a model of starfish predation on coral" ''IMA J. Math. Appl. in Medicine and Biol.'' , '''4''' (1987) pp. 207–214 {{MR|910187}} {{ZBL|}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> "Acanthaster and the coral reef: A theoretical perspective" R. Bradbury (ed.) , ''Lecture Notes Biomath.'' , '''88''' , Springer (1990)</td></tr><tr><td valign="top">[a5]</td> <td valign="top"> P. Breen, T.A. Caros, J.B. Foster, E.A. Stewart, "Changes in subtidal community structure associated with British Columbia sea otter transplants" ''Marine Eco.'' , '''7''' (1982) pp. 13–20</td></tr><tr><td valign="top">[a6]</td> <td valign="top"> J.T. Bryant, "The regulation of snowshoe hare feeding behaviour during winter by plant anti-herbivore chemistry" K. Myers (ed.) C.D. McInness (ed.) , ''Proc. World Lagomorph Conf.'' , Guelph Univ. Press (1979)</td></tr><tr><td valign="top">[a7]</td> <td valign="top"> N. Elredge, "Time frames, the evolution of punctuated equilibria" , Princeton Univ. Press (1989)</td></tr><tr><td valign="top">[a8]</td> <td valign="top"> C.S. Elton, "The ecology of invasion by animals and plants" , Methuen (1958)</td></tr><tr><td valign="top">[a9]</td> <td valign="top"> R. Endean, "Acanthaster planci infestations of reefs of the Great Barrier Reef" , ''Proc. Third Internat. Coral Reef Symp.'' , '''1''' (1977) pp. 185–191</td></tr><tr><td valign="top">[a10]</td> <td valign="top"> G.F. Gause, "The struggle for existence" , Williams and Wilkins (1934)</td></tr><tr><td valign="top">[a11]</td> <td valign="top"> G.F. Gause, A.A. Witt, "Behaviour of mixed populations and the problem of natural selection" ''Amer. Nat.'' , '''69''' (1935) pp. 596–609</td></tr><tr><td valign="top">[a12]</td> <td valign="top"> M.E. Gilpin, "Do hares eat lynx?" ''Amer. Nat.'' , '''107''' (1973) pp. 727–730</td></tr><tr><td valign="top">[a13]</td> <td valign="top"> M. Gilpin, F.J. Ayala, "Global models of growth and competition" ''Proc. Nat. Acad. Sci.'' , '''70''' (1973) pp. 3590–3593 {{MR|}} {{ZBL|0272.92016}} </td></tr><tr><td valign="top">[a14]</td> <td valign="top"> J.L. Harper, "The population biology of plants" , Acad. Press (1977)</td></tr><tr><td valign="top">[a15]</td> <td valign="top"> B. Hassard, N.D. Kazarinoff, Y.-H. Wan, "Theory and applications of Hopf bifurcations" , ''London Math. Soc. Lecture Notes'' , '''41''' , Cambridge Univ. Press (1981) {{MR|0603442}} {{ZBL|}} </td></tr><tr><td valign="top">[a16]</td> <td valign="top"> G.E. Hutchinson, "An introduction to population biology" , Yale Univ. Press (1978)</td></tr><tr><td valign="top">[a17]</td> <td valign="top"> L.B. Keith, "Wildlife's ten-year cycle" , Univ. Wisconsin Press (1963)</td></tr><tr><td valign="top">[a18]</td> <td valign="top"> K. Mann, "Kelp, sea urchins and predators: a review of strong interactions in rocky subtidal systems of eastern Canada 1970–1980" ''Netherl. J. Sea Research'' , '''16''' (1982) pp. 414–423</td></tr><tr><td valign="top">[a19]</td> <td valign="top"> D.F. Rhoades, "Offensive-defensive interactions between herbivores and plants: their relevance in herbivore population dynamics and ecological theory" ''Amer. Nat.'' , '''125''' (1985) pp. 205–223</td></tr><tr><td valign="top">[a20]</td> <td valign="top"> R.E. Ricklefs, "Ecology" , Chiron Press (1979) (Edition: Second)</td></tr><tr><td valign="top">[a21]</td> <td valign="top"> P.L. Antonelli, R. Bradbury, "Volterra–Hamilton models in the ecology and evolution of colonial organisms" , ''Ser. Math. Biol. and Medicine'' , World Sci. (1996) {{MR|}} {{ZBL|0930.92031}} </td></tr></table>

Latest revision as of 16:46, 1 July 2020

There are two kinds of models in mathematical ecology, broadly speaking. There are, on the one hand, models of strategic type, which are based on empirical formulas and use computer simulation techniques. These are popular among ecologists because they fit the data extremely well and are highly predictive in particular cases, say a wheat field in Saskatchewan or a sheep herd in New Zealand. But, in fact, they tell next to nothing about the underlying ecology. On the other hand, there are dynamical models, which often involve ordinary differential equations, but may use stochastic differential equations, difference equations, integral equations, or diffusion reaction equations. These models encode postulates about ecological mechanisms into the equations. As a rule, these do not predict as well as strategic models do, because of the constraints imposed by these postulates. But it is through the use of dynamical models that tentative explanations can be found and eventual consensus reached, so that more general, improved, strategic models can be designed for ecosystem management. Below, for the sake of brevity only ordinary differential equation models are considered.

Growth of a single population.

Let $N ( t )$ denote the total number, or density, of a population $\Sigma$ at a fixed location and time. Assume that $N ( t )$ is continuous in the time $t$. The Hutchinson postulates [a16] are:

1) $d N / d t = f ( N )$, $f$ sufficiently differentiable;

2) $N \equiv 0$ implies $d N / d t \equiv 0$;

3) $N ( t )$ is bounded between zero and a fixed positive constant $C$, for all time.

Given the Hutchinson postulates for a population $\Sigma$, it follows that the ordinary differential equation

\begin{equation} \tag{a1} \frac { d N } { d t } = \lambda N \left( 1 - \frac { N } { K } \right) , \end{equation}

for which

\begin{equation} \tag{a2} N ( t ) = \frac { K } { 1 + b e ^ { - \lambda t } } \end{equation}

is the general solution, is the simplest growth law. It is called the logistic equation.

The parameter $K$, called the carrying capacity for $\Sigma$, obviously satisfies $0 < K \leq C$. The parameter $\lambda > 0$ is called the intrinsic growth rate. Of the four types of shapes specified for (a1) by $b < 0$, $b = 0$, $0 < b \leq 1$, $b > 1$, only the last is $S$-shaped (i.e. its graph has an inflection point).

Suppose that $\Sigma$ satisfies only (a1) and (a2); then, denoting $n ( t ) = N ( t ) - N_ {*}$, where $f ( N_{ *} ) = 0$ (i.e. $N _{*}$ is a steady-state), Taylor expansion around $N _{*}$ gives

\begin{equation*} \frac { d N } { d t } = \frac { d n } { d t } = f ( N ) = \end{equation*}

\begin{equation*} = f ( N_{ * } ) + f ^ { \prime } ( N_{ * } ) n + \frac { f ^ { \prime \prime } ( N_{ * } ) } { 2 } n ^ { 2 } + \ldots, \end{equation*}

where the prime denotes differentiation with respect to $N$. For $n ( t )$ small in absolute value, $d n / d t$ is well approximated by $f ^ { \prime } ( N_{*} ) n$. Therefore, $n ( t )$ increases with time if $f ^ { \prime } ( N _{*} ) > 0$, and decreases if $f ^ { \prime } ( N_{*} ) < 0$. In the former case, $N _{*}$ is an unstable steady-state while, in the latter case, $N _{*}$ is a stable steady-state.

For the logistic special case, $N_{*} = 0$ or $N_* = K$ are the only possible steady-states, the former being unstable and the latter stable.

The logistic differential equation (a1) is the simplest description of a population with limited resources, the limitation being provided by the negative coefficient of the quadratic term. The equation first arose in the work of P. Verhulst (1838) and later in the demographic research of R. Pearl and L. Reed in the 1920s. It was subsequently used to provide a dynamic model of malaria in humans by Sir Ronald Ross, but has perhaps a more basic role in ecology than in epidemiology.

Growth dynamics in a competitive community.

Several species living in the same locality must forage for food and seek nesting sites in a field or stream, etc. These populations may or may not affect one another.

Suppose that $n$ species comprise a community $\Sigma$ in which there are no inter-specific interactions. This ecosystem can be modeled by

\begin{equation} \tag{a3} \frac { d N ^ { i } } { d t } = \lambda _ { ( i ) } N ^ { i } \left( 1 - \frac { N ^ { i } } { K _ { ( i ) } } \right) , \quad i = 1 , \ldots , n, \end{equation}

where $N ^ { i }$ denotes the total number or density of the $i$th species in $\Sigma$. This system has $2 ^ { n }$ steady-states, but only $( K _ { ( 1 ) } , \dots , K _ { ( n ) } )$ is stable. The equations (a3) describe non-competition.

Now suppose there is competition for food items, etc. How does one describe this? G.F. Gause and A.A. Witt answered this for a $2$-species community ($n = 2$) with [a11]

\begin{equation} \tag{a4} \left\{ \begin{array}{l}{ \frac { d N ^ { 1 } } { d t } = \lambda _ { ( 1 ) } N ^ { 1 } \left( 1 - \frac { N ^ { 1 } } { K _ { ( 1 ) } } - \delta _ { ( 1 ) } \frac { N ^ { 2 } } { K _ { ( 1 ) } } \right), }\\{ \frac { d N ^ { 2 } } { d t } = \lambda _ { ( 2 ) } N ^ { 2 } \left( 1 - \frac { N ^ { 2 } } { K _ { ( 2 ) } } - \delta _ { ( 2 ) } \frac { N ^ { 1 } } { K _ { ( 2 ) } } \right). }\end{array} \right. \end{equation}

Here, all $\lambda$, $K$ and $\delta$ are positive. This system has exactly one positive equilibrium $( N _ { * } ^ { 1 } , N _ { * } ^ { 2 } )$, given by

\begin{equation} \tag{a5} \left\{ \begin{array}{l}{ N _ { * } ^ { 1 } = \frac { K _ { ( 1 ) } - \delta _ { ( 1 ) } K _ { ( 2 ) } } { 1 - \delta _ { ( 1 ) } \delta _ { ( 2 ) } }, }\\{ N _ { * } ^ { 2 } = \frac { K _ { ( 2 ) } - \delta _ { ( 2 ) } K _ { ( 1 ) } } { 1 - \delta _ { ( 1 ) } \delta _ { ( 2 ) } }. }\end{array} \right. \end{equation}

If both numerators and denominators are positive, then $( N _ { * } ^ { 1 } , N _ { * } ^ { 2 } )$ in (a5) is stable. If they are both negative, (a5) is unstable. This is easily proved by using the stability Ansatz: the eigenvalues of the Jacobian of the right-hand side of a system

\begin{equation*} \frac { d N ^ { i } } { d t } = f ^ { i } ( N ^ { 1 } , \ldots , N ^ { n } ) , \quad i = 1 , \dots , n, \end{equation*}

evaluated at a steady-state $( N _ { * } ^ { 1 } , \ldots , N _ { * } ^ { n } )$, must have negative real part for stability. If any of these is positive, an unstable case results.

In the question of survival for the two populations in Gause–Witt competition (a4), (a5), there are four cases to consider:

A) If $\delta _{( 1 )} > K _ { ( 1 ) } / K _ { ( 2 ) }$ and $\delta_{( 2 )} > K _ { ( 2 ) } / K _ { ( 1 ) }$, then (a5) is unstable, with survival depending on the initial proportions of $N ^ { 1 }$ and $N ^ { 2 }$.

B) If $\delta _{( 1 )} > K _ { ( 1 ) } / K _ { ( 2 ) }$ and $\delta _ { ( 2 ) } < K _ { ( 2 ) } / K _ { ( 1 ) }$, then (a5) is unstable, and the first species will be eliminated.

C) If $\delta _ { ( 1 ) } < K _ { ( 1 ) } / K _ { ( 2 ) }$ and $\delta_{( 2 )} > K _ { ( 2 ) } / K _ { ( 1 ) }$, then (a5) is unstable, and the second species will be eliminated.

D) If $\delta _ { ( 1 ) } < K _ { ( 1 ) } / K _ { ( 2 ) }$ and $\delta _ { ( 2 ) } < K _ { ( 2 ) } / K _ { ( 1 ) }$, then (a5) is stable.

Therefore, only in case D), called incomplete competition, can both species coexist. This case translates as some geometrical separation of the two species, where the more vulnerable one has a refuge it can retreat to, or some resource available that the otherwise better adapted competitor cannot use [a16].

Experiments performed by Gause on Paramecium [a10] verified the outcomes A)–D) qualitatively. Thus, the Gause–Witt equations imply that complete competitors cannot coexist. This is the famous principle of competitive exclusion, a corner-stone of mathematical ecology. There are variants and generalizations of this principle; see, e.g., [a2]. This generality underscores the fundamental importance of that principle. Indeed, biologists claim that competition between species has profound evolutionary consequences [a7].

Three-species interactions: a general model applicable to several different ecosystems.

One of the great benefits of dynamical models is their tendency to be applicable in more than one ecological situation. This is partly because they are framed in precise mathematical terms encoding a list of specific postulates and assumptions, but also because in their qualitative behaviour lies the essence of their application. An illustration of this is the example given below, of a model known to encompass three different ecosystems. The model exhibits switching between multiple steady-states and stable periodic solutions (i.e. stable limit cycles) induced by predation of one species on another. In its full generality, the system (a9) models predation on a herbivore which in turn feeds on a plant species. The limit cycle behaviour described is not induced by time-lags, as in the classical Lotka–Volterra predator-prey model (with predator devastating the prey population to the extent that there is not enough prey for the much larger predator population, which then crashes, resulting in the prey population coming back full circle). Rather, the mechanism is aggregation, caused by spawning or feeding behaviour of the predator, conditioned by environmental constraints in some cases (e.g. cyclones, drought, nutrient enrichment, etc.). Furthermore, one must always prove that a periodic solution, topologically a circle, is stable, in the sense that there is a solid torus $T$ in phase-space whose centre is the cycle and having the property that any solution with initial conditions in $T$ will converge onto that cycle as $t \rightarrow + \infty$. The methods of Hopf bifurcation provide the necessary tools for this analysis [a15].

The logistic growth equation with exponential parameter $a > 0$,

\begin{equation} \tag{a6} \frac { d N } { d t } = \lambda N \left( 1 - \left( \frac { N } { K } \right) ^ {a } \right), \end{equation}

was introduced to explain certain data on Drosophila in [a13]. The case $a > 1$ indicates greater self-inhibition while the converse is true for $a < 1$. Similarly, the dynamical model

\begin{equation} \tag{a7} \frac { d F } { d t } = - \varepsilon F ( 1 - \gamma F ^ { p } ), \end{equation}

where $\varepsilon > 0$, $\gamma > 0$ and $p \in ( 1 / 2,3 / 2 )$, was introduced to explain crown-of-thorns starfish (Acanthaster planci) aggregation on coral reefs [a1], [a21]. The term $\gamma F ^ { p }$ is called the cooperative term. If $p < 1$, then the variable coefficient of $F ^ { 2 }$ in (a7) is relatively large for small values of $F$. This results in increased cooperation, and the reverse is true for $p > 1$. The parameter $\gamma$ is the coefficient of aggregation. It also serves as Hopf bifurcation parameter in (a8) and (a9), where Hopf's method can be used to prove the existence of small amplitude-stable periodic solutions (i.e. stable limit cycles), [a15]. Note that $p$ is fixed in a model, unlike $\gamma$, which is a free parameter. Rather, $p$ is an indicator of fecundity or genetically determined potential for reproduction. The role of $p$ and $\gamma$ in (a8) and (a9) is investigated below.

Consider the ordinary differential equations

\begin{equation} \tag{a8} \left. \begin{cases} { \frac { d N } { d t } = N ( - 2 \alpha N - \delta F + \lambda ), } \\ { \frac { d F } { d t } = F ( 2 \beta N + \gamma F ^ { p } - \varepsilon ), } \end{cases} \right. \end{equation}

where $p$ is taken slightly less than $1$. The constants $\alpha$, $\delta$, $\beta$, $\gamma$ can be given a precisely defined chemical interpretation based on the concepts of the Volterra production variable and on the Rhoades allometric plant response mechanism [a19], [a21]; $N$ denotes the density of plant modular units (e.g. leaves); [a14]. $F$ is the density of the herbivore population in the same locality. The system (a8) is a model in the theory of optimal defense of plants against herbivores, [a19].

Use of Hopf bifurcation theory and the Hassard code BIFOR2 show the existence of a stable periodic solution (i.e. limit cycle) of small amplitude [a1], [a15]. One may also show that the amplitude can be large [a3]. It is also possible to show that the period of the cycle is longer for plants which use the metabolically expensive chemical defense (e.g. oaks), as opposed to plants (e.g. herbaceous) which do not. This explains both the 9–10 year cycle of the oak caterpillar and the 3–4 cycle of voles and lemmings which eat herbaceous plants. The model requires $p \ll 1$ so that the herbivore must not only have highly aggregative behaviour, but must be highly fecund.

An interesting application of the chemically mediated plant/herbivore system (a8) is to the lynx-snowshoe hare (Lepus americanus) cycle in the Arctic (cf. also Canadian lynx data; Canadian lynx series). $N$ denotes the modular unit density for the plant and $F$ the hare density. The large reproductive potential of the $F$-population is interpreted as $p \ll 1$. It is known that the plants which hares eat are chemically defended and that this has a strong negative effect on the hare population. It was discovered in the field that the hare population cycles both with and without the presence of lynx [a8], [a17], [a6]! The three species extension of (a8), which incorporates the lynx, is

\begin{equation} \tag{a9} \begin{cases} { l } { \frac { d N } { d t } = N ( - 2 \alpha N - \delta F + \lambda ) }, \\ { \frac { d F } { d t } = F ( 2 \beta N + \gamma F ^ { p } - \varepsilon - \mu _ { 1 } L ) }, \\ { \frac { d L } { d t } = \mu _ { 2 } L F - \nu L }, \end{cases} \end{equation}

where all constants are non-negative. For convenience one sets $\alpha = \beta$, but this has no biological significance. However, if one also sets $\mu _ { 2 } = \gamma$ and rewrites the third equation as

\begin{equation} \tag{a10} \frac { d L } { d t } = \gamma L ( F - \xi ) , \quad \xi = \frac { \nu } { \gamma }, \end{equation}

the assumption $\mu _ { 2 } = \gamma$ implies that the predator $L$ is getting more food value out of its kill, all other things being equal, as $\gamma$ increases. This model shows that the high reproductivity of the Arctic hare population ($p \ll 1$) drives a stable periodic cycle whose period increases with increasing amounts of defensive compounds in the plant tissues. Also, the $F$-population will cycle without the lynx and so the lynx-hare cycle is driven by the hare's food quality, with the lynx population going along piggy-back style. This model is an improvement over the time-lag model, [a20], [a12].

A model of Acanthaster planci predation on corals of the Great Barrier Reef is provided by (a8), but without the chemical interpretation for the $N$-population, which in this case is coral. The starfish population is highly fecund and aggregates, causing outbreaks with a 12–15 year period [a4]. Thus, $p \ll 1$ will generate a bifurcation from the positive equilibrium of (a8) to a stable periodic cycle triggered by increasing $\gamma$ beyond a certain critical Hopf value determined by the coefficients in (a8). The extended system (a9) can be used to discuss the claim of marine biologist R. Endean that the giant conch, C. tritonis, which preys on adult Acanthaster plani, may be a keystone predator on the Great Barrier Reef [a9]. Such a conception excludes any limit cycle behaviour, a priori, and is essentially a steady-state theory. Assuming that C. tritonis gains when starfish aggregate (i.e. $\gamma$ increases) and that $p \ll 1$, the model (a9) predicts Hopf bifurcation from a steady-state to a stable limit cycle of moderate amplitude. Consequently, C. tritonis must also cycle synchronously (i.e. piggy-back). However, there is no evidence for regular conch fluctuations in this case. Yet, if A. planci were neither highly fecund nor aggregative, then $p \geq 1$ would have to be used in (a9) and the result would be a steady-state (perhaps several). That is, giant triton would be a keystone predator, similar to the role of sea otters in the Western Canadian sea urchin-kelp system discussed below.

On the west coast of North America, red sea urchins (S. franciscanus) feed on kelps in large aggregates and exist in at least two possible steady-states: at very low density within kelp beds ($\delta \neq 0$) in the presence of sea otters; or at high density outside kelp beds ($\delta \approx 0$) in the absence of sea otters $( L _ { 0 } \approx 0 )$ [a5]. If $L _ { 0 } \approx 0$, then $F _ { 0 }$ is relatively large, as is $N_ 0 $. The system (a9) has a unique positive equilibrium for $\lambda - \delta \xi > 0$ and $2 \beta N _ { 0 } + \gamma \xi ^ { p } - \varepsilon > 0$, $p > 1$. It is

\begin{equation} \tag{a11} N _ { 0 } = \frac { \lambda - \delta \xi } { 2 \alpha } , L _ { 0 } = \frac { 2 \beta N _ { 0 } + \gamma \xi ^ { p } - \varepsilon } { \mu _ { 1 } } , F _ { 0 } = \xi. \end{equation}

In the case where $\delta \approx 0$, the system reduces to one with steady-state: ($N _ { 0 } = \lambda / ( 2 \alpha )$, $L _ { 0 } = 0$, $F _ { 0 } = \xi$), with $p > 1$. It is known from field data that the steady-state can rapidly switch and depends only on the presence or absence of sea otters. The otter is a keystone predator causing rapid switching in the red sea urchin population.

The above model also applies to the lobster-sea urchin-kelp system of the Eastern Canadian coast. In this case the lobsters play the keystone predator role, [a18].

References

[a1] "Mathematical essays on growth and the emergence of form" P.L. Antonelli (ed.) , Univ. Alberta Press (1985) MR0826076 Zbl 0572.00026
[a2] P. Antonelli, R. Bradbury, X. Lin, "On Hutchinson's competition equations and their homogenization: A higher-order principle of competitive exclusion" Ecol. Modelling , 60 (1992) pp. 309–320
[a3] P.L. Antonelli, K.D. Fuller, N.D. Kazarinoff, "A study of large amplitude periodic solutions in a model of starfish predation on coral" IMA J. Math. Appl. in Medicine and Biol. , 4 (1987) pp. 207–214 MR910187
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How to Cite This Entry:
Mathematical ecology. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Mathematical_ecology&oldid=15284
This article was adapted from an original article by P.L. Antonelli (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article