Long-range interactions and evolutionary stability in a predator-prey system

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Long-range interactions and evolutionary stability in a predator-prey system

Erik M. Rauch and Yaneer Bar-Yam

Phys. Rev. E 73, 020903(R) – Published 27 February 2006

Abstract

Evolving ecosystems often are dominated by spatially local dynamics, but many also include long-range transport that mixes spatially separated groups. The existence of such mixing may be of critical importance since research shows spatial separation may be responsible for long-term stability of predator-prey systems. Complete mixing results in rapid global extinction, while spatial systems achive long term stability due to an inhomogeneous spatial pattern of local extinctions. We consider the robustness of a generic evolving predator-prey or host-pathogen model to long-range mixing and find a transition to global extinction at nontrivial values implying that even if significant mixing already exists, a small amount of additional mixing may cause extinction. Our results are relevant to the global mixing of species due to human intervention and to global transport of infectious disease.

Received 31 December 2004

DOI:https://doi.org/10.1103/PhysRevE.73.020903

Authors & Affiliations

Erik M. Rauch^1,2 and Yaneer Bar-Yam²

¹MIT Computer Science and Artificial Intelligence Laboratory, 32 Vassar St., Cambridge, Massachusetts 02139, USA
²New England Complex Systems Institute, 24 Mt. Auburn St., Cambridge, Massachusetts 02138, USA

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Vol. 73, Iss. 2 — February 2006

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Images

Figure 1
(Color) Snapshots of the lattice for the evolving pathogen-host model. Twenty generations elapse between each frame. Prey populations by themselves are shown as green and prey being consumed by predators are colored depending on their value of the predator reproduction rate $τ$ , as shown in the legend. In this and all subsequent figures, the system has settled to a steady state value of $τ$ . We use an $L \times L$ square lattice with periodic boundary conditions and a von Neumann neighborhood (north, south, east, and west neighbors). In this figure, the lattice size $L = 100$ , prey depletion rate $v = 0.2$ , and prey reproduction rate $g = 0.05$ .Reuse & Permissions
Figure 2
(a) Evolutionary stability as a function of lattice size $L$ . Each point represents the probability $p_{es}$ that the predator and prey will coexist for at least 100 000 generations, obtained from 10 runs. Any time substantially longer than a single generation will give similar results as there is a sharp transition between populations that survive for times that are exponential in system size and those that are not more than linear. There is a transition at a size $L_{es}$ . For the left curve (circles), the prey reproduction rate $g = 0.2$ . For the right curve (× symbols), $g = 0.05$ . The depletion rate $v = 0.2$ . (b) The minimum lattice size for evolutionary stability $L_{es}$ as a function of the average total population $S$ of an overexploiting strain over the course of its existence. Results are for all combinations of $v = 0.1, 0.2, 0.4$ and $g = 0.05, 0.1, 0.2$ . The line shows the fit $L_{es} = 0.1 S + 30$ $(r^{2} = 0.97)$ .Reuse & Permissions
Figure 3
Evolutionary stability on a two-dimensional Small-World network. (a) The probability $p_{es}$ that predator and prey coexist for 100 000 generations, as a function of $p$ , averaged over 11 runs, for $g = 0.05$ (circles) and $g = 0.2$ (squares) (depletion rate $v = 0.2$ , lattice size $L = 250$ ). Note the logarithmic scale of $p$ . We identify the point of transition to instability $p_{c}$ as the density such that for all $p > p_{c}$ , $p_{es} < 1 ∕ 2$ . For comparison, the average path length $l (p)$ between nodes is plotted as a fraction of $l (0)$ (dashed line, same scale). For comparison, the dotted line shows $l (1) ∕ l (0)$ , that is, the value for a random network. (b) The evolutionarily stable reproduction rate $τ_{es}$ as a function of $p$ on a two-dimensional (2D) Small-World network, averaged over the last 200 generations of 10 runs of 100 000 generations. $τ_{es}$ is also plotted for values of $p$ for which the predators go extinct (shaded region); the average of the last 200 generations before extinction is plotted. (c) As (b), but for $g = 0.05$ .Reuse & Permissions
Figure 4
(a) The critical density $p_{c}$ as a function of the average patch area $a_{p}$ . The line is the least-squares fit $p_{c} \sim a_{p}^{- 1.5}$ $(r^{2} = 0.94)$ . (b) The length scale of the crossover between “small-world” and “large-world” scaling of distances at the critical density $p_{c}$ , as a function of the average patch diameter $\sqrt{a_{p}}$ , i.e., $ξ (p_{c}) = l n (4 π p_{c} L^{2}) ∕ 2 \sqrt{π p_{c}}$ . The line shows the fit $ξ (p_{c}) = 2.1 \sqrt{a_{p}}$ $(r^{2} = 0.97)$ . Please note the logarithmic horizontal scale.Reuse & Permissions

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