Slow evolution towards generalism in a model of variable dietary range

Abstract

Species sharing a habitat will co-evolve to make use of the available resources, as consumption is modulated by competition and negative feedback loops between consumers and resources. The dietary range of a given species determines the resources it has access to and thus the other species with which it competes. A narrow dietary range avoids competition at the cost of over-reliance on a small selection of resources; conversely a wide dietary range provides more alternatives but also more chance of competition with other species. Here, we investigate the evolution of dietary range within a mathematical model of niche formation. We find highly path dependent co-evolution dynamics characterised by long-lived quasi-stable states. Ultimately, stochastic effects drive the evolution of generalist diets, as we uncover in our analysis and simulations.

Figures (8)

• Figure 1: A: The competition between two species at positions $x_1$ and $x_2$ is given by the overlap in their resource preference distributions. B: The competition kernel for this type of indirect competition is always positive definite, as it is the convolution of the resource preference distribution with itself. Thus, species are not expected to form in the model. C,D: This can be demonstrated with a numerical solution to the deterministic equations which slowly tends towards the homogeneous steady state, even from an initially heterogeneous distribution. E,F: The demographic noise in stochastic simulations, however, causes patterns to persist, interpreted as species. The pattern arising can be predicted from analytical analysis of the deterministic equations. C and E show the consumer distributions and D and F show the resource distributions.
• Figure 2: The power spectrum $\left<|\tilde{c}_k|^2\right>$ for various (integer) values of $\frac{kL}{2\pi}$ between zero and 40 and values for the dietary range, $w$, between zero and one. For each value of $w$, the power spectrum was normalised by dividing all values of the spectrum by the power of the highest peak. The parameter values for the system were $\alpha=2\times10^4$, $\beta=1$, $\delta=2$, $\gamma=0.01$, $\kappa=1\times10^5$ and $L=1$. In A, $\mu=10^{-7}$ and in B, $\mu=10^{-5}$.
• Figure 3: The evolution of a system containing individuals that can evolve their preferred resources (moving along the resource axis $x$ (100 bins)) and whose dietary range is $w=0.2$. A: Initially there are five consumers in each bin along $x$. Five roughly equally spaced species form rapidly and despite noise causing various speciation and extinction events, the system recovers to a state with five species equally spaced along $x$. B: Initially there were 10 equally spaced species in this system. Once an extinction event happens (which is very soon in the simulation) the system is unlikely to recover to the initial state, instead evolving rapidly to the state with five equally spaced species. The parameters of this system were $\alpha=2\times10^4$, $\beta=1$, $\delta=2$, $\gamma=0.01$, $\kappa=1\times10^5$, $\mu=10^{-5}$.
• Figure 4: The dominant pattern-forming mode for different combinations of $w_2$ and $w_2-w_1$. In the case where $w_1\rightarrow w_2$ the results from the analysis without evolvable dietary range are recovered. As the width of the window of permitted dietary ranges expands, the higher-$k$ modes become less dominant until the first mode dominates (interpreted as a single species). The parameter values for this system were $\alpha=2\times10^4$, $\beta=1$, $\delta=2$, $\gamma=0.01$ and $\kappa=1\times10^5$. In A $\mu_x=\mu_w=10^{-7}$. In B $\mu_x=\mu_w=10^{-5}$. We see that greater noise leads to loss of some of the finer structure of this plot and the higher modes (i.e., the coexistence of a greater number of species). The vector $\mathbf{w}$ contained 100 entries, giving a $101\times101$ matrix to invert in each case (see Appendix \ref{['sec:CoarseGrainingDietaryRange']} for details). Note that the area below the black line $w_2=w_2-w_1$ (i.e., $w_1=0$) is not meaningful for this system. The entire area to the right of the region shown has dominant mode equal to one.
• Figure 5: The system in both upper and lower panels is in the same state at $t=0$: five species with dietary range $w=0.2$, equally spaced along the resource axis, $x$ (100 bins), at the moment the centrally positioned species goes extinct. The upper panels show the evolution of this system when $0<w<1$ (50 bins), and the lower panels show the evolution when $0.19<w<0.21$ (5 bins): an unrestricted and a restricted evolution of dietary range scenario. The upper panels show an increase in dietary range on average to satisfy Equation \ref{['eq:FlatConvolutionCondition']} with four heterogeneous species (note the different widths of the "columns" in the right hand plot). The lower panels show that the only way to satisfy the condition given the restrictions on $w$ is with five species, equally spaced in $x$. The parameters of these systems were $\alpha=2\times10^4$, $\beta=1$, $\delta=2$, $\gamma=0.01$, $\kappa=1\times10^5$, $\mu_x=\mu_w=10^{-5}$.