Survival and extinction for a contact process with a density-dependent birth rate
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arXiv
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| Format: | Preprint |
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2024
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| author | Köppl, Jonas Lanchier, Nicolas Mercer, Max |
| author_facet | Köppl, Jonas Lanchier, Nicolas Mercer, Max |
| contents | To study later spatial evolutionary games based on the multitype contact process, we first focus in this paper on the conditions for survival/extinction in the presence of only one strategy, in which case our model consists of a variant of the contact process with a density-dependent birth rate. The players are located on the $d$-dimensional integer lattice, with natural birth rate $λ$ and natural death rate one. The process also depends on a payoff $a_{11} = a$ modeling the effects of the players on each other: while players always die at rate one, the rate at which they give birth is given by $λ$ times the exponential of $a$ times the fraction of occupied sites in their neighborhood. In particular, the birth rate increases with the local density when $a > 0$, in which case the payoff $a$ models mutual cooperation, whereas the birth rate decreases with the local density when $a < 0$, in which case the payoff $a$ models intraspecific competition. Using standard coupling arguments to compare the process with the basic contact process (the particular case $a = 0$), we prove that, for all payoffs $a$, there is a phase transition from extinction to survival in the direction of $λ$. Using various block constructions, we also prove that, for all birth rates $λ$, there is a phase transition in the direction of $a$. This last result is in sharp contrast with the behavior of the nonspatial deterministic mean-field model in which the stability of the extinction state only depends on $λ$. This underlines the importance of space (local interactions) and stochasticity in our model. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09944 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Survival and extinction for a contact process with a density-dependent birth rate Köppl, Jonas Lanchier, Nicolas Mercer, Max Probability 60K35, 91A22 To study later spatial evolutionary games based on the multitype contact process, we first focus in this paper on the conditions for survival/extinction in the presence of only one strategy, in which case our model consists of a variant of the contact process with a density-dependent birth rate. The players are located on the $d$-dimensional integer lattice, with natural birth rate $λ$ and natural death rate one. The process also depends on a payoff $a_{11} = a$ modeling the effects of the players on each other: while players always die at rate one, the rate at which they give birth is given by $λ$ times the exponential of $a$ times the fraction of occupied sites in their neighborhood. In particular, the birth rate increases with the local density when $a > 0$, in which case the payoff $a$ models mutual cooperation, whereas the birth rate decreases with the local density when $a < 0$, in which case the payoff $a$ models intraspecific competition. Using standard coupling arguments to compare the process with the basic contact process (the particular case $a = 0$), we prove that, for all payoffs $a$, there is a phase transition from extinction to survival in the direction of $λ$. Using various block constructions, we also prove that, for all birth rates $λ$, there is a phase transition in the direction of $a$. This last result is in sharp contrast with the behavior of the nonspatial deterministic mean-field model in which the stability of the extinction state only depends on $λ$. This underlines the importance of space (local interactions) and stochasticity in our model. |
| title | Survival and extinction for a contact process with a density-dependent birth rate |
| topic | Probability 60K35, 91A22 |
| url | https://arxiv.org/abs/2404.09944 |