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Coexistence for a population model with forest fire epidemics

Luis Fredes, Amitai Linker, Daniel Remenik

Abstract

We investigate the effect on survival and coexistence of introducing forest fire epidemics to a certain two-species competition model. The model is an extension of the one introduced by Durrett and Remenik [DR09], who studied a discrete time particle system running on a random 3-regular graph where occupied sites grow until they become sufficiently dense so that an epidemic wipes out large clusters. In our extension we let two species affected by independent epidemics compete for space, and we allow the epidemic to attack not only giant clusters, but also clusters of smaller order. Our main results show that, for the two-type model, there are explicit parameter regions where either one species dominates or there is coexistence; this contrasts with the behavior of the model without epidemics, where the fitter species always dominates. We also discuss the survival and extinction regimes for the model with a single species. In both cases we prove convergence to explicit dynamical systems; simulations suggest that their orbits present chaotic behavior.

Coexistence for a population model with forest fire epidemics

Abstract

We investigate the effect on survival and coexistence of introducing forest fire epidemics to a certain two-species competition model. The model is an extension of the one introduced by Durrett and Remenik [DR09], who studied a discrete time particle system running on a random 3-regular graph where occupied sites grow until they become sufficiently dense so that an epidemic wipes out large clusters. In our extension we let two species affected by independent epidemics compete for space, and we allow the epidemic to attack not only giant clusters, but also clusters of smaller order. Our main results show that, for the two-type model, there are explicit parameter regions where either one species dominates or there is coexistence; this contrasts with the behavior of the model without epidemics, where the fitter species always dominates. We also discuss the survival and extinction regimes for the model with a single species. In both cases we prove convergence to explicit dynamical systems; simulations suggest that their orbits present chaotic behavior.

Paper Structure

This paper contains 18 sections, 22 theorems, 115 equations, 4 figures.

Table of Contents

  1. Introduction and main results
  2. The multi-type moth model on a random 3-regular graph
  3. Coexistence and domination for the two-type MMM
  4. The limiting dynamical system
  5. Derivation of the limit
  6. Approximation result
  7. Phase diagrams
  8. The one-type system and bifurcation cascades
  9. The two type dynamical system
  10. Connection with the particle system
  11. Proof of the approximation result
  12. Proof of the main result
  13. Interior-recurrent sets
  14. Preliminaries
  15. Proof of Theorem \ref{['extcoex']}(ii)
  16. ...and 3 more sections

Key Result

Theorem 1.1

Consider the two-species MMM on a random 3-regular graph satisfying eq:alphaconv and $1<\phi_{1}<\phi_{2}$ and let $\underline{\alpha}_N=\min\{\alpha_N(1),\alpha_N(2)\}$. Then there are constants $c_1,c_2,c_1',c_2'>0$ such that the following holds: For any fixed $0<l_1<u_1<1$ and $0<l_2<u_2<1$ there for all $N$ and any $\rho_{0}^{N}\space(1)\in[l_1,u_1]$, $\rho_{0}^{N}\space(2)\in[l_2,u_2]$ (that

Figures (4)

  • Figure 1: Left: Bifurcation diagram in $\beta$ for ${\sf DS}(h)$ with $\alpha=0.1$, showing the orbits of the system between iterations 900 and 1000 in the vertical direction for different values of $\beta$. Our simulations suggest that cascades appear for all $\alpha \in (0,1)$. Right: Simulation of the evolution of the MM for $\alpha=0.1$ and different values of $\beta$, from iteration 900 to 1000. Here $N\in \{20000,40000,100000\}$ (depending on $\beta$).
  • Figure 2: Approximate phase diagram of ${\sf DS}(h)$. The transition between extinction and survival is justified by \ref{['prop8']}, while the one governing the appearance of bifurcation cascades (dashed line) is based on simulations.
  • Figure 3: Summary of the domination and coexistence regimes for the MMM, for $\alpha\StrLen{1}[\x] =\alpha\StrLen{2}[\x] =0$ on the left and $\alpha\StrLen{1}[\x] =\alpha\StrLen{2}[\x] = 0.1$ on the right. The white (resp. black) dashed regions represent the domination regime of type 1 over type 2 (resp. type 2 over type 1), and the solid gray regions correspond roughly to the coexistence regime (plotted based on their asymptotic behavior: as $\phi_{2}\rightarrow \infty$, $\phi_{1}$ grows as $\sqrt{\phi_{2}\log(\phi_{2})}$).
  • Figure 4: Bifurcation diagrams for type 1 (blue) and type 2 (black), with $\alpha\StrLen{1}[\x] =0.01$ and $\alpha\StrLen{2}[\x] =0.2$. On the left, with $\beta_1=1.99\log(2)$, type 1 goes from a stable fixed point to extinction as $\beta_2$ increases. On the right, with $\beta_1=2$, there is coexistence for large $\beta_2$; note how the chaotic behavior of the type 2 species is reflected on type 1 as well.

Theorems & Definitions (45)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Proposition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Theorem 2.7
  • ...and 35 more