Analyzed the data: YBZ GY NMS. Contributed reagents/materials/analysis tools: YBZ GY NMS. Wrote the paper: YBZ GY NMS.
The authors have declared that no competing interests exist.
The persistence of a spatially structured population is determined by the rate of dispersal among habitat patches. If the local dynamic at the subpopulation level is extinction-prone, the system viability is maximal at intermediate connectivity where recolonization is allowed, but full synchronization that enables correlated extinction is forbidden. Here we developed and used an algorithm for agent-based simulations in order to study the persistence of a stochastic metapopulation. The effect of noise is shown to be dramatic, and the dynamics of the spatial population differs substantially from the predictions of deterministic models. This has been validated for the stochastic versions of the logistic map, the Ricker map and the Nicholson-Bailey host-parasitoid system. To analyze the possibility of extinction, previous studies were focused on the attractiveness (Lyapunov exponent) of stable solutions and the structure of their basin of attraction (dependence on initial population size). Our results suggest that these features are of secondary importance in the presence of stochasticity. Instead, optimal sustainability is achieved when decoherence is maximal. Individual-based simulations of metapopulations of different sizes, dimensions and noise types, show that the system's lifetime peaks when it displays checkerboard spatial patterns. This conclusion is supported by the results of a recently published
No one can produce all his needs by himself. Personal autarky poses a serious danger of collapse in cases of illness, drought, etc. Trade reduces the impact of local catastrophes, thus increasing economic stability. However, the recent series of econo-crises revealed that globalization induces coherence among markets and jeopardizes their sustainability against global failures. Economists try to identify the optimal tariff that balances between the dangers of autarky and the risk of correlated failure. The same problem appears in ecosystems with a population divided among local habitat patches. “Optimal tariff” is translated to optimal migration rate: how should one manipulate connectivity among patches in order to achieve maximum sustainability? Recolonization of habitats that undergo extinction is essential for survival, yet a too strong dispersal leads to coherence and correlated extinction. Here we use individual-based models in order to find the optimal migration rate. We show that this optimum appears when the the system takes a spatial “checkerboard” pattern that maximizes the decoherence. The insights gleaned allow for improved policies for conservation of endangered species (optimizing the effect of corridors, predicting the impact of habitat fragmentation) and, on the other hand, eradication campaigns (like vaccination or pest control).
In recent years, many studies in the field of biodiversity maintenance were focused on spatially structured populations
The population of an isolated patch is usually unstable, as demographic and environmental fluctuations may drive the colony to extinction. Migration among subpopulations allows recolonization of vacant habitat patches (turnover events) and reduces the risk of correlated extinction
The situation becomes much more complicated if the local dynamics of a large, well-mixed population is also extinction-prone. In such a case, strong dispersal, which is equivalent to patch merging, increases spatial coherence and leads to global extinction. Many recent experiments on predator-prey
The dynamic is illustrated by a cartoon (upper panel). Intra-patch logistic growth is followed by a migration step; the graphs indicate the average lifetime
There is a substantial literature on the two edges of the bell shape: the extinction transition that takes place as migration becomes too small
Our main result is the identification of the conditions for maximum sustainability. Surprisingly, it turns out that the optimal point for the stochastic system has nothing to do with the stability properties of the deterministic (noise-free) dynamics. Instead, it always appears when the spatial system arranges itself in a
Along this paper we deal solely with demographic stochasticity. However, it should be emphasized that our results hold in the presence of other types of noise, like the environmental stochasticity considered by
First let us present the numerical technique used in order to study the effect of demographic stochasticity on the sustainability of a spatially segregated population. We demonstrate this technique for the logistic system; the generalization of this method to any other dynamics is presented in the
We consider a metapopulation with
Each of the
Indeed,
The deterministic limit of the stochastic-logistic system corresponds, thus, to the paradigmatic model of diffusively coupled logistic maps, considered already in the context of population dynamics
Note that this deterministic model
Does this fact indicate that the peak should be
Here the deterministic (
The time evolution of total population (
These individual particle simulations show that for realistic values of
The fractal basin boundary and the dependence of the asymptotic behavior on initial conditions have already been pointed out by Adler
(A): The orbit diagram for the deterministic system [Eq. (7)] with
The orbit diagram (A) presented together with the Lyapunov exponent
The orbit diagram (A) is presented only for the narrow range of parameters where the deterministic map [Eq. (11)] supports periodic orbits; in that region the persistence curve (C) of the individual-based dynamics [Eq. (12)] admits its maximum. In Panel (B), the population on patch 1 (P1) and 2 (P2) is given at different times for
In the orbit diagram (A), the black region corresponds to the UUDD orbits, and the green to the UDUD. In the UDUD region the persistence curve (C) of the individual-based dynamics admits its maximum. In Panel (B), the population sizes on patch 1–4 is given at different times for
In the NB case we can show the orbit diagram for only a limited region of the phase space, where generic initial conditions converge to an attractive orbit, a phenomenon first pointed out by Adler
In both the Ricker and the Nicholson-Baily maps, the system achieves maximum sustainability in the “UDUD” region, similarly to the case of the logistic map. Here too, the value of the Lyapunov exponent turns out to be irrelevant for determining the maximum sustainabilty point.
The exceptional stability of the checkerboard pattern has to do with the fact that in this state the decoherence among neighboring habitat patches is maximal. To understand this we briefly review some elements of previous studies.
A generic mechanism that leads to sustainability in spatially structured populations has been discovered recently
In order to grasp the essence of the stabilizing mechanism, let us look at
An illustration of a two-patch system (up), where the intra-patch dynamics on each follows the deterministic Nicholson-Bailey model, as described by Eqs. 11. Here
Stable orbits, thus, may appear due to the presence of noise. The role of noise is to perturb the system from its fully synchronized phase. Once this perturbation happens, the differences between patches are amplified by the underlying unstable dynamic. This yields an effective decoherence between patches and, as a result, the dynamic stabilizes.
Based on this insight, we suggest that the optimal persistence is always achieved at the point of maximum decoherence. The basic unit is a two-patch model in the “up-down” phase, and the whole system should be tiled with these dominoes in a checkerboard array that allows for maximum rescue effect. As demonstrated in
Average time to extinction (lower right, arb. units) vs. migration rate for one-dimensional array of 16 patches (green) and for
As shown above, the very same result holds for different dynamics that acquire stability through spatial structure, like the Ricker map considered by
The results of the
All these considerations fail when the number of individuals per site becomes extremely large (the system follows the deterministic dynamics and the stability of an orbit is governed by the Lyapunov exponent) or small (where synchronization is no longer important and migration always helps). These limits are discussed below.
Within the general framework suggested by Earn, Levin and Rohani
The checkerboard solution manifests itself even if the topology of the system does not allow for “perfect” partition, as in the case of an odd number of patches or an imperfect lattice. As demonstrated in the
The numerical procedure used along this work is a generic individual-based generalization of the deterministic approach for coupled map lattice
A very similar island model has been used by Hamilton and May
On the contrary, here we consider the case where in the deterministic limit, the dynamics are unstable (chaotic or otherwise extinction-prone), and thus stochasticity induces fluctuations, and their interference with the spatial structure plays a crucial role in the persistence of the population.
We assume a population dynamics with nonoverlapping generations, where any generation involves two consecutive steps. The first step involves the “local reaction” (birth, death, competition etc., at which any patch is affected only by the local population), and the second is the density independent “migration” (dispersal) step, where individuals are allowed, with a certain probability, to leave their local community and migrate to another patch. No “dispersal cost” is introduced, so any emigrant reaches its chosen destination.
In the reaction step, the number of agents on a patch at the
Both maps are chaotic, and after a while the system reaches population levels that are very close to zero. The deterministic formalism has no problems with that: since
After the reaction step, a migration step takes place. In the deterministic limit, a fraction
Another example we consider here is the non-chaotic (yet extinction-prone) Nicholson-Bailey
Along this work we have used three procedures in order to estimate
If the system reaches extinction on reasonable timescales, we can simply average over the time to extinction obtained from repeated runs of the simulation with random initial conditions and different histories. The result is
To implement this method, one assumes that the population fluctuates, more or less normally, around
Here we did not use the stochastic simulations at all. Instead, it is assumed that the demographic fluctuations effectively kicked the system away from the stable orbit, into a random location (picked with uniform distribution among all possible states) in the phase space. This chosen point becomes a new “initial condition” that flows, in the deterministic limit, back to the attractive orbit along a transient trajectory. The chance of extinction during this transient is proportional to the minimum over time of the total population,
In
The value of
There are two extreme cases of a too large and too small noise, where the checkerboard strategy fails. In the weak noise limit (that corresponds to the large
The other limit, that of large noise, appears when
The optimal migration rate (the one that yields the maximal time to extinction)
First, in the main only demographic stochasticity was considered. Here we show that the same results hold for a system subject to both environmental and demographic stochasticity. Second, in the main part of the paper only the average time to extinction was presented. Here we show the whole probability function and confirm that it is exponential distribution.
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As the migration parameter D increases, the fractal map showing Xm for any possible initial condition of a two patch system is changes (lower right). In the lower left panel the orbit diagram is updated for any given D. the upper panel shows the average of Xm over all possible initial states; it yields the bell-shape with the peak at the optimal sustainability point, as explained in the text.
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The individual based Ricker dynamic is simulated on a 10×10 lattice (periodic boundary conditions), with a “defect” (inaccessible sits, dark blue) in the middle. The movie present consecutive snapshots of the density of particles, color coded as indicated by the color bar, at the optimal migration point. One realizes that the system reaches the checkerboard state, with a single moving imperfection localized close to the defect.
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