Stochastic models in population biology and their deterministic analogs

We introduce a class of stochastic population models based on “patch dynamics.” The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean-field theories, which are generally valid as the patch size becomes very large. These models may be used to formulate a broad range of biological processes in both spatial and nonspatial contexts. Here, we concentrate on two-species competition. We present both a mathematical analysis of the patch model, in which we derive the precise form of the competition mean-field equations (and their first-order corrections in the nonspatial case), and simulation results. These mean-field equations differ, in some important ways, from those which are normally written down on phenomenological grounds. Our general conclusion is that mean-field theory is more robust for spatial models than for a single isolated patch. This is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by interpatch diffusion. However, discrete effects due to modest patch sizes lead to striking deviations from mean-field theory even in a spatial setting.

Figure 1

A comparison of theory to simulation for a single patch containing a single species. The upper panel shows the time evolution of the population density $\varphi \left(t\right)=⟨n⟩∕N$, where $n$ is the number of individuals and $N$ is the size of the patch. The lower panel shows the time evolution of the variance, $v\left(t\right)=(⟨n^2⟩-{⟨n⟩}^2)∕N^2$, of the population. The subscript $A$ refers to the fact that the individuals belong to species $A$. In this case the patch size has the relatively large value of 100 and we see that theory and simulation are in good agreement. See Table for specific parameter values used in Figs. 1, 2.

Figure 2

The same as Fig. 1 but with the patch size reduced to 10. In this case the mean population density falls to zero due to fluctuation induced extinctions. The true variance first exceeds the large $N$ prediction and then falls steeply below due to the growing number of realizations which have become extinct.

Figure 3

Comparison of theory to simulation for a patch containing two species $A$ and $B$. Initially each species has a density of one-quarter of the total patch capacity. The upper panel shows the time evolution of the densities of $A$ and $B$ [$\varphi \left(t\right)$ and $\psi \left(t\right)$, respectively], while the lower panel shows the time evolution of the variance of $A$ compared with the large-$N$ theory. In this case the $A$ and $B$ individuals have identical birth, death, and competition parameters, and the interspecific competition is set to zero. The patch size has the large value of 400. See Table for specific parameter values used in Figs. 3, 4, 5.

Figure 4

The same as Fig. 3 but with a patch size of 200. Note that although the mean-field theory still works fairly well, the large-$N$ prediction for the variance is very poor. Even for such a large patch, the fluctuations are increasing steadily with time.

Figure 5

A similar scenario to Fig. 3, with the addition of weak interspecific interactions (one-fifth of the strength of the intraspecific interactions, with $A$ outcompeting $B$). Here, the patch size is 400, and the densities are fairly well approximated by mean-field theory, although we see a slow decline in the $B$ population. However, the fluctuations are again increasing with time and are not well described by the large-$N$ theory for large times.

Figure 6

Comparison of mean-field theory (smooth lines) and simulation (erratic lines) for two species $A$ and $B$ in a spatial setting in which initially the $A$ individuals occupy the left half of the system and the $B$ individuals the right half. The upper (lower) panel shows early (late) times. Here, the patch size has the relatively large value of 100. $A$ outcompetes $B$ (meaning that $c_{21}>c_{12}$), but has a higher death rate, and so is invaded by $B$. See Table for specific parameter values used in Figs. 6, 7, 8, 9.

Figure 7

The same as Fig. 6 but here the patch size is reduced from 100 to the relatively small value of 10. Note that mean-field theory is still in fairly good agreement with the high density $B$ population, but shows significant deviation for the stochastically weakened $A$ population.

Figure 8

A similar scenario to Fig. 6, but now $A$ and $B$ have identical death rates and yet $B$ has less intraspecific competition than $A$. In this example $A$ invades $B$. The patch size here has the relatively large value of 50.

Figure 9

The same as Fig. 8 except that the patch size is reduced from 50 to 10. Note that the invasion of $A$ into $B$ is severely slowed due to the stochastic weakening of $A$.