Three interacting particle systems arising from biology

We study three types of interacting particle systems arising from biology.

In Chapter 2, we study an evolutionary game in which a producer at $ x $ gives birth at rate 1 to an offspring sent to a randomly chosen point in $x + {\cal N} _ c$, while a cheater at $x$ gives birth at rate $\lambda> 1$ times the fraction of producers in $x + {\cal N}_d$ and sends its offspring to a randomly chosen point in $x + {\cal N}_c$. We first study this game on the $d$-dimensional torus $\TL$ with ${\cal N}_d = \TL$ and ${\cal N} _ c$ = the $2d$ nearest neighbors. If we let $L \to \infty$ then $t \to \infty$ the fraction of producers converges to $1/\lambda$. In $d \ge 3$ the limiting finite dimensional distributions converge as $t \to \infty$ to the voter model equilibrium with density $1/\lambda$ . We next reformulate the system as an evolutionary game with ``birth-death'' updating and take ${\cal N}_c = {\cal N}_d = \cal N}$. Using results for voter model perturbations we show that in $d = 3$ with ${\cal N} =$ the six nearest neighbors, the density of producers converges to $(2/\lambda)-0.5$ for $4/3 < \lambda < 4$. Producers take over the system when $\lambda < 4/3$ and die out when $\lambda >4$. In $d=2$ with ${\cal N} = [-c\sqrt{\log N},c\sqrt{\log N}]^2$ there are similar phase transitions, with coexistence occurring when $(1+2\theta)/(1+\theta) < \lambda < (1+2\theta) / \theta$ where $\theta = (e^{3/(\pi c^2)}-1)/2$.

In Chapter 3, we formulate a nonhomogeneous spatial model of the competition between forest and savanna. In work with a variety of co-authors, Staver and Levin have argued that savanna and forest coexist as alternative stable states with discontinuous changes in density of trees at the boundary. We prove this claim and that coexistence occurs for a time that is exponential in the size of the system, and that after an initial transient, boundaries between the alternative equilibria remain stable.

In Chapter 4, we study a two-level contact process. We think of fleas living on a species of animals. We let the animals follow the law of a supercritical contact process in $\mathbb{ Z }^d$ with parameter $ \lambda $. The contact process acts as the random environment where neutral symbionts (``fleas'') grow. The fleas give birth at rate $\mu$ when they are living on a host animal, and the fleas die at rate $\delta$ when they do not have a host animal. Using a block construction, we show that if the contact process is supercritical, the fleas survive with a positive probability for any sufficiently large $\mu$. The main result of this chapter is the complete convergence theorem of the model.