Mathematical ecology

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

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