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A system of two species, one feeding on the other (cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]]). A variety of mathematical approaches is used when modelling a predator-prey system, since there are many factors that can influence its evolution, e.g. predation-mediated coexistence, the size of habitat, hierarchical ranking, voracity and fertility of species, competition, inhomogeneity with respect to the age structure, latent, infection and incubation lags, seasonal changes, space diffusion, pollution, spatial environment heterogeneity, finite acceptance time for external signals, carrying capacity, permanence (persistence), etc. Isolating those factors which have to be taken into consideration and neglecting the others, one obtains different mathematical predator-prey models, described by different types of equations: ordinary, partial or functional differential equations, deterministic or stochastic, discrete or continuous etc.
 
A system of two species, one feeding on the other (cf. [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]], [[#References|[a5]]], [[#References|[a6]]]). A variety of mathematical approaches is used when modelling a predator-prey system, since there are many factors that can influence its evolution, e.g. predation-mediated coexistence, the size of habitat, hierarchical ranking, voracity and fertility of species, competition, inhomogeneity with respect to the age structure, latent, infection and incubation lags, seasonal changes, space diffusion, pollution, spatial environment heterogeneity, finite acceptance time for external signals, carrying capacity, permanence (persistence), etc. Isolating those factors which have to be taken into consideration and neglecting the others, one obtains different mathematical predator-prey models, described by different types of equations: ordinary, partial or functional differential equations, deterministic or stochastic, discrete or continuous etc.
  
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This model, which takes into account only intrinsic phenomena (voracity and fertility), has the form
 
This model, which takes into account only intrinsic phenomena (voracity and fertility), has the form
  
<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/p/p110/p110200/p1102001.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\left .
 +
 
 +
\begin{array}{c}
 +
{\dot{x} } ( t ) = [ a _ {1} - a _ {2} y ( t ) ] x ( t ) ,  \\
 +
{\dot{y} } ( t ) = [ a _ {3} x ( t ) - a _ {4} ] y ( t ) .  \\
 +
\end{array}
  
Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p1102002.png" /> is the number of preys, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p1102003.png" /> is the number of predators, and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p1102004.png" /> are positive constants (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p1102005.png" /> is the rate of birth of preys, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p1102006.png" /> is the rate of death of predators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p1102007.png" /> is a measure of susceptibility of preys to predation, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p1102008.png" /> is the predatory ability). The system (a1) has a unique non-zero [[Equilibrium position|equilibrium position]], which is a stable centre. At the same time, the solutions of (a1) are not structurally stable with respect to disturbance of initial conditions (cf. [[Rough system|Rough system]]).
+
\right \}
 +
$$
 +
 
 +
Here, $  x ( t ) $
 +
is the number of preys, $  y ( t ) $
 +
is the number of predators, and the $  a _ {i} $
 +
are positive constants ( $  a _ {1} $
 +
is the rate of birth of preys, $  a _ {4} $
 +
is the rate of death of predators, $  a _ {2} $
 +
is a measure of susceptibility of preys to predation, and $  a _ {3} $
 +
is the predatory ability). The system (a1) has a unique non-zero [[Equilibrium position|equilibrium position]], which is a stable centre. At the same time, the solutions of (a1) are not structurally stable with respect to disturbance of initial conditions (cf. [[Rough system|Rough system]]).
  
 
==Models with intraspecific strife.==
 
==Models with intraspecific strife.==
 
Within the restrictive domain of quadratic differential equations, those which include competition as well as predation should be somewhat more realistic. An example of a model with competition inside preys is given by:
 
Within the restrictive domain of quadratic differential equations, those which include competition as well as predation should be somewhat more realistic. An example of a model with competition inside preys is 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/p/p110/p110200/p1102009.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
\left .
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020010.png" /> characterizes competition inside preys. Under biologically reasonable assumptions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020011.png" />, and (a2) has a unique positive equilibrium which is an asymptotically-stable focus or node (cf. [[Focus|Focus]]; [[Node|Node]]; [[Asymptotically-stable solution|Asymptotically-stable solution]]).
+
\begin{array}{c}
 +
{\dot{x} } ( t ) = [ a _ {1} - a _ {2} y ( t ) - a _ {5} x ( t ) ] x ( t ) ,  \\
 +
{\dot{y} } ( t ) = [ a _ {3} x ( t ) - a _ {4} ] y ( t ) ,  \\
 +
\end{array}
 +
 
 +
\right \}
 +
$$
 +
 
 +
where  $  a _ {5} > 0 $
 +
characterizes competition inside preys. Under biologically reasonable assumptions, $  a _ {1} a _ {3} > a _ {5} a _ {4} $,  
 +
and (a2) has a unique positive equilibrium which is an asymptotically-stable focus or node (cf. [[Focus|Focus]]; [[Node|Node]]; [[Asymptotically-stable solution|Asymptotically-stable solution]]).
  
 
==Kolmogorov predator-prey model.==
 
==Kolmogorov predator-prey model.==
 
This model can be written in the form
 
This model can be written in the form
  
<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/p/p110/p110200/p11020012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
+
$$ \tag{a3 }
 +
\left .
  
under appropriate conditions on the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020014.png" /> and their derivatives, reflecting a number of real phenomena, such as satiation. Node, focus and limit cycles (cf. [[Limit cycle|Limit cycle]]) are among the possible behaviours of (a3).
+
\begin{array}{c}
 +
{\dot{x} } ( t ) = a ( x,y ) x ( t ) ,  \\
 +
{\dot{y} } ( t ) = b ( x,y ) y ( t ) ,  \\
 +
\end{array}
 +
 
 +
\right \}
 +
$$
 +
 
 +
under appropriate conditions on the functions $  a ( x,y ) $,
 +
$  b ( x,y ) $
 +
and their derivatives, reflecting a number of real phenomena, such as satiation. Node, focus and limit cycles (cf. [[Limit cycle|Limit cycle]]) are among the possible behaviours of (a3).
  
 
==Delay effects.==
 
==Delay effects.==
 
To achieve some degree of realism, delay effects have to be taken into account. Moreover, many phenomena, like instability, oscillation and periodic change, cannot be explained without implementing delays into the model. To account for delay of feeding on reproduction, V. Volterra introduced the equations
 
To achieve some degree of realism, delay effects have to be taken into account. Moreover, many phenomena, like instability, oscillation and periodic change, cannot be explained without implementing delays into the model. To account for delay of feeding on reproduction, V. Volterra introduced the 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/p/p110/p110200/p11020015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
$$ \tag{a4 }
 +
\left .  
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020017.png" /> are finite positive continuous functions. The system (a4) leads to a stable aperiodic oscillation. Other types of integro-differential [[#References|[a7]]] and, more general, functional-differential equations [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]] are widely used to develop predator-prey models of moderate mathematical complexity while not sacrificing biological realism.
+
\begin{array}{c}
 +
{ {x _ {1} } dot } ( t ) = \left [ a _ {1} - \int\limits _ { 0 } ^  \infty  {x _ {2} ( t - s ) K _ {1} ( s ) }  {ds } \right ] x _ {1} ( t ) ,  \\
 +
{ {x _ {2} } dot } ( t ) = \left [ - a _ {4} - \int\limits _ { 0 } ^  \infty  {x _ {1} ( t - s ) K _ {2} ( s ) }  {ds } \right ] x _ {2} ( t ) ,  \\
 +
\end{array}
 +
 +
\right \}
 +
$$
 +
 
 +
where  $  K _ {1} $
 +
and $  K _ {2} $
 +
are finite positive continuous functions. The system (a4) leads to a stable aperiodic oscillation. Other types of integro-differential [[#References|[a7]]] and, more general, functional-differential equations [[#References|[a8]]], [[#References|[a9]]], [[#References|[a10]]] are widely used to develop predator-prey models of moderate mathematical complexity while not sacrificing biological realism.
  
 
==Discrete-time models.==
 
==Discrete-time models.==
 
The discrete version of (a3) is:
 
The discrete version of (a3) 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/p/p110/p110200/p11020018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>
+
$$ \tag{a5 }
 +
\left .
 +
 
 +
\begin{array}{c}
 +
x _ {t + 1 }  = x _ {t} a ( x _ {t} ,y _ {t} ) ,  \\
 +
y _ {t + 1 }  = y _ {t} a ( x _ {t} ,y _ {t} ) ,  \\
 +
\end{array}
 +
 
 +
\right \}
 +
$$
  
where the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020020.png" /> relate the predator-influenced reproductive efficiency of the prey and the searching of the predator, respectively. Even a simple version of (a5) can exhibit rich dynamics, from stability to [[Chaos|chaos]]. For example, the equation (with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020021.png" /> a constant parameter)
+
where the functions $  a $
 +
and $  b $
 +
relate the predator-influenced reproductive efficiency of the prey and the searching of the predator, respectively. Even a simple version of (a5) can exhibit rich dynamics, from stability to [[Chaos|chaos]]. For example, the equation (with $  a $
 +
a constant parameter)
  
<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/p/p110/p110200/p11020022.png" /></td> </tr></table>
+
$$
 +
x _ {t + 1 }  = x _ {t} [ a ( 1 - x _ {t} ) - y _ {t} ] ,  y _ {t + 1 }  = {
 +
\frac{x _ {t} y _ {t} }{0.31 }
 +
}
 +
$$
  
has an asymptotically-stable fixed point for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020023.png" />, an invariant circle for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020024.png" />, and an attracting set for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020025.png" />.
+
has an asymptotically-stable fixed point for $  a \in [ 0,2.6 ] $,  
 +
an invariant circle for $  a \in ( 2.6,3.44 ] $,  
 +
and an attracting set for $  a > 3.44 $.
  
 
==Predator-prey system with spatial inhomogeneity.==
 
==Predator-prey system with spatial inhomogeneity.==
Such a system can be modelled by partial differential equations. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020027.png" /> denote the prey and predator densities at time <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020028.png" /> at the space point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020029.png" />. Then, under the assumption that all dispersal occurs solely by simple diffusion processes, the predator-prey model has the form of reaction-diffusion equations (cf. [[Reaction-diffusion equation|Reaction-diffusion equation]]):
+
Such a system can be modelled by partial differential equations. Let $  x ( t,r ) $,
 +
$  y ( t,r ) $
 +
denote the prey and predator densities at time $  t $
 +
at the space point $  r $.  
 +
Then, under the assumption that all dispersal occurs solely by simple diffusion processes, the predator-prey model has the form of reaction-diffusion equations (cf. [[Reaction-diffusion equation|Reaction-diffusion equation]]):
 +
 
 +
$$ \tag{a6 }
 +
\left .
 +
 
 +
\begin{array}{c}
 +
{
 +
\frac{\partial  x }{\partial  t }
 +
} = \sigma _ {1} \Delta x + a ( x,y ) x,  \\
 +
{
 +
\frac{\partial  y }{\partial  t }
 +
} = \sigma _ {2} \Delta y + b ( x,y ) y,  \\
 +
\end{array}
  
<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/p/p110/p110200/p11020030.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>
+
\right \}
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020031.png" /> are the diffusion rates and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020032.png" /> is the [[Laplace operator|Laplace operator]]. If the diffusion rates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020033.png" /> are increasing, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020034.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020035.png" /> become spatially homogeneous for large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020036.png" />, i.e. the behaviour of this predator-prey system can be described by (a3). Diffusion in (a6) can generate instability, just opposite to its usual interpretation as a smoothing mechanism. In order to incorporate different realistic effects, it is often necessary to introduce time delays into the governing equations (a6) as well.
+
where $  \sigma _ {i} $
 +
are the diffusion rates and $  \Delta $
 +
is the [[Laplace operator|Laplace operator]]. If the diffusion rates $  \sigma _ {i} $
 +
are increasing, then $  x $,  
 +
$  y $
 +
become spatially homogeneous for large $  t $,  
 +
i.e. the behaviour of this predator-prey system can be described by (a3). Diffusion in (a6) can generate instability, just opposite to its usual interpretation as a smoothing mechanism. In order to incorporate different realistic effects, it is often necessary to introduce time delays into the governing equations (a6) as well.
  
 
==Predator-prey models with uncertainties of various origins.==
 
==Predator-prey models with uncertainties of various origins.==
These can be described by stochastic equations, e.g. of the form (a3), (a5) or (a6), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110200/p11020038.png" /> are stochastic processes (cf. also [[Stochastic differential equation|Stochastic differential equation]]).
+
These can be described by stochastic equations, e.g. of the form (a3), (a5) or (a6), where $  a $
 +
and $  b $
 +
are stochastic processes (cf. also [[Stochastic differential equation|Stochastic differential equation]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V. Volterra, "Théorie mathématique de la lutte pour la vie" , Gauthier-Villars (1931) {{MR|1189803}} {{ZBL|0002.04202}} {{ZBL|57.0466.02}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.J. Lotka, "Elements of physical biology" , Williams and Witkins (1925) {{MR|0094263}} {{MR|1521010}} {{ZBL|51.0416.06}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.N. Kolmogoroff, "Sulla theoria di Volterra della lotta per l'esistenza" ''Giorn. Inst. Ital. Attuari'' , '''7''' (1936) pp. 74–80</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> V.A. Kostitzin, "Mathematical biology" , Harrap (1939) {{MR|}} {{ZBL|0025.20302}} {{ZBL|65.1361.01}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Smith, "Models in ecology" , Cambridge Univ. Press (1974) {{MR|}} {{ZBL|0312.92001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J. Murray, "Mathematical biology" , Springer (1989) {{MR|1007836}} {{ZBL|0682.92001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> C.M. Cushing, "Integro-differential equations and delay models in population dynamics" , ''Lecture Notes in Biomathematics'' , '''20''' , Springer (1977) {{MR|496838}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> V.B. Kolmanovskii, A.D. Myshkis, "Applied theory of functional differential equations" , Kluwer Acad. Publ. (1992) (In Russian) {{MR|1256486}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> Y. Kuang, "Delay differential equations with applications in population dynamics" , Acad. Press (1993) {{MR|1218880}} {{ZBL|0777.34002}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> K. Gopalsamy, "Equations of mathematical ecology" , Kluwer Acad. Publ. (1992)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> V. Volterra, "Théorie mathématique de la lutte pour la vie" , Gauthier-Villars (1931) {{MR|1189803}} {{ZBL|0002.04202}} {{ZBL|57.0466.02}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> A.J. Lotka, "Elements of physical biology" , Williams and Witkins (1925) {{MR|0094263}} {{MR|1521010}} {{ZBL|51.0416.06}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A.N. Kolmogoroff, "Sulla theoria di Volterra della lotta per l'esistenza" ''Giorn. Inst. Ital. Attuari'' , '''7''' (1936) pp. 74–80</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> V.A. Kostitzin, "Mathematical biology" , Harrap (1939) {{MR|}} {{ZBL|0025.20302}} {{ZBL|65.1361.01}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> M. Smith, "Models in ecology" , Cambridge Univ. Press (1974) {{MR|}} {{ZBL|0312.92001}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> J. Murray, "Mathematical biology" , Springer (1989) {{MR|1007836}} {{ZBL|0682.92001}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> C.M. Cushing, "Integro-differential equations and delay models in population dynamics" , ''Lecture Notes in Biomathematics'' , '''20''' , Springer (1977) {{MR|496838}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> V.B. Kolmanovskii, A.D. Myshkis, "Applied theory of functional differential equations" , Kluwer Acad. Publ. (1992) (In Russian) {{MR|1256486}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> Y. Kuang, "Delay differential equations with applications in population dynamics" , Acad. Press (1993) {{MR|1218880}} {{ZBL|0777.34002}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> K. Gopalsamy, "Equations of mathematical ecology" , Kluwer Acad. Publ. (1992)</TD></TR></table>

Latest revision as of 14:54, 7 June 2020


A system of two species, one feeding on the other (cf. [a1], [a2], [a3], [a4], [a5], [a6]). A variety of mathematical approaches is used when modelling a predator-prey system, since there are many factors that can influence its evolution, e.g. predation-mediated coexistence, the size of habitat, hierarchical ranking, voracity and fertility of species, competition, inhomogeneity with respect to the age structure, latent, infection and incubation lags, seasonal changes, space diffusion, pollution, spatial environment heterogeneity, finite acceptance time for external signals, carrying capacity, permanence (persistence), etc. Isolating those factors which have to be taken into consideration and neglecting the others, one obtains different mathematical predator-prey models, described by different types of equations: ordinary, partial or functional differential equations, deterministic or stochastic, discrete or continuous etc.

Lotka–Volterra model.

This model, which takes into account only intrinsic phenomena (voracity and fertility), has the form

$$ \tag{a1 } \left . \begin{array}{c} {\dot{x} } ( t ) = [ a _ {1} - a _ {2} y ( t ) ] x ( t ) , \\ {\dot{y} } ( t ) = [ a _ {3} x ( t ) - a _ {4} ] y ( t ) . \\ \end{array} \right \} $$

Here, $ x ( t ) $ is the number of preys, $ y ( t ) $ is the number of predators, and the $ a _ {i} $ are positive constants ( $ a _ {1} $ is the rate of birth of preys, $ a _ {4} $ is the rate of death of predators, $ a _ {2} $ is a measure of susceptibility of preys to predation, and $ a _ {3} $ is the predatory ability). The system (a1) has a unique non-zero equilibrium position, which is a stable centre. At the same time, the solutions of (a1) are not structurally stable with respect to disturbance of initial conditions (cf. Rough system).

Models with intraspecific strife.

Within the restrictive domain of quadratic differential equations, those which include competition as well as predation should be somewhat more realistic. An example of a model with competition inside preys is given by:

$$ \tag{a2 } \left . \begin{array}{c} {\dot{x} } ( t ) = [ a _ {1} - a _ {2} y ( t ) - a _ {5} x ( t ) ] x ( t ) , \\ {\dot{y} } ( t ) = [ a _ {3} x ( t ) - a _ {4} ] y ( t ) , \\ \end{array} \right \} $$

where $ a _ {5} > 0 $ characterizes competition inside preys. Under biologically reasonable assumptions, $ a _ {1} a _ {3} > a _ {5} a _ {4} $, and (a2) has a unique positive equilibrium which is an asymptotically-stable focus or node (cf. Focus; Node; Asymptotically-stable solution).

Kolmogorov predator-prey model.

This model can be written in the form

$$ \tag{a3 } \left . \begin{array}{c} {\dot{x} } ( t ) = a ( x,y ) x ( t ) , \\ {\dot{y} } ( t ) = b ( x,y ) y ( t ) , \\ \end{array} \right \} $$

under appropriate conditions on the functions $ a ( x,y ) $, $ b ( x,y ) $ and their derivatives, reflecting a number of real phenomena, such as satiation. Node, focus and limit cycles (cf. Limit cycle) are among the possible behaviours of (a3).

Delay effects.

To achieve some degree of realism, delay effects have to be taken into account. Moreover, many phenomena, like instability, oscillation and periodic change, cannot be explained without implementing delays into the model. To account for delay of feeding on reproduction, V. Volterra introduced the equations

$$ \tag{a4 } \left . \begin{array}{c} { {x _ {1} } dot } ( t ) = \left [ a _ {1} - \int\limits _ { 0 } ^ \infty {x _ {2} ( t - s ) K _ {1} ( s ) } {ds } \right ] x _ {1} ( t ) , \\ { {x _ {2} } dot } ( t ) = \left [ - a _ {4} - \int\limits _ { 0 } ^ \infty {x _ {1} ( t - s ) K _ {2} ( s ) } {ds } \right ] x _ {2} ( t ) , \\ \end{array} \right \} $$

where $ K _ {1} $ and $ K _ {2} $ are finite positive continuous functions. The system (a4) leads to a stable aperiodic oscillation. Other types of integro-differential [a7] and, more general, functional-differential equations [a8], [a9], [a10] are widely used to develop predator-prey models of moderate mathematical complexity while not sacrificing biological realism.

Discrete-time models.

The discrete version of (a3) is:

$$ \tag{a5 } \left . \begin{array}{c} x _ {t + 1 } = x _ {t} a ( x _ {t} ,y _ {t} ) , \\ y _ {t + 1 } = y _ {t} a ( x _ {t} ,y _ {t} ) , \\ \end{array} \right \} $$

where the functions $ a $ and $ b $ relate the predator-influenced reproductive efficiency of the prey and the searching of the predator, respectively. Even a simple version of (a5) can exhibit rich dynamics, from stability to chaos. For example, the equation (with $ a $ a constant parameter)

$$ x _ {t + 1 } = x _ {t} [ a ( 1 - x _ {t} ) - y _ {t} ] , y _ {t + 1 } = { \frac{x _ {t} y _ {t} }{0.31 } } $$

has an asymptotically-stable fixed point for $ a \in [ 0,2.6 ] $, an invariant circle for $ a \in ( 2.6,3.44 ] $, and an attracting set for $ a > 3.44 $.

Predator-prey system with spatial inhomogeneity.

Such a system can be modelled by partial differential equations. Let $ x ( t,r ) $, $ y ( t,r ) $ denote the prey and predator densities at time $ t $ at the space point $ r $. Then, under the assumption that all dispersal occurs solely by simple diffusion processes, the predator-prey model has the form of reaction-diffusion equations (cf. Reaction-diffusion equation):

$$ \tag{a6 } \left . \begin{array}{c} { \frac{\partial x }{\partial t } } = \sigma _ {1} \Delta x + a ( x,y ) x, \\ { \frac{\partial y }{\partial t } } = \sigma _ {2} \Delta y + b ( x,y ) y, \\ \end{array} \right \} $$

where $ \sigma _ {i} $ are the diffusion rates and $ \Delta $ is the Laplace operator. If the diffusion rates $ \sigma _ {i} $ are increasing, then $ x $, $ y $ become spatially homogeneous for large $ t $, i.e. the behaviour of this predator-prey system can be described by (a3). Diffusion in (a6) can generate instability, just opposite to its usual interpretation as a smoothing mechanism. In order to incorporate different realistic effects, it is often necessary to introduce time delays into the governing equations (a6) as well.

Predator-prey models with uncertainties of various origins.

These can be described by stochastic equations, e.g. of the form (a3), (a5) or (a6), where $ a $ and $ b $ are stochastic processes (cf. also Stochastic differential equation).

References

[a1] V. Volterra, "Théorie mathématique de la lutte pour la vie" , Gauthier-Villars (1931) MR1189803 Zbl 0002.04202 Zbl 57.0466.02
[a2] A.J. Lotka, "Elements of physical biology" , Williams and Witkins (1925) MR0094263 MR1521010 Zbl 51.0416.06
[a3] A.N. Kolmogoroff, "Sulla theoria di Volterra della lotta per l'esistenza" Giorn. Inst. Ital. Attuari , 7 (1936) pp. 74–80
[a4] V.A. Kostitzin, "Mathematical biology" , Harrap (1939) Zbl 0025.20302 Zbl 65.1361.01
[a5] M. Smith, "Models in ecology" , Cambridge Univ. Press (1974) Zbl 0312.92001
[a6] J. Murray, "Mathematical biology" , Springer (1989) MR1007836 Zbl 0682.92001
[a7] C.M. Cushing, "Integro-differential equations and delay models in population dynamics" , Lecture Notes in Biomathematics , 20 , Springer (1977) MR496838
[a8] V.B. Kolmanovskii, A.D. Myshkis, "Applied theory of functional differential equations" , Kluwer Acad. Publ. (1992) (In Russian) MR1256486
[a9] Y. Kuang, "Delay differential equations with applications in population dynamics" , Acad. Press (1993) MR1218880 Zbl 0777.34002
[a10] K. Gopalsamy, "Equations of mathematical ecology" , Kluwer Acad. Publ. (1992)
How to Cite This Entry:
Predator-prey system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Predator-prey_system&oldid=24537
This article was adapted from an original article by V. Kolmanovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article