Modeling species dispersal with occupancy urn models

Abstract

Models for species dispersal make various simplifications to facilitate analysis, such as ignoring spatial correlations or assuming equal probability of colonization among all sites within a dispersal neighborhood. Here we introduce a variation of the basic contact process (BCP) which allows us to separate the number of offspring produced from the neighborhood size, which are confounded in the original BCP. We then use classical results arising from probability models involving placing balls in urns to study our modified BCP, obtaining bounds for the critical value of the survival probability needed for the population to persist. We also use the probability urn calculations with a local-dispersal mean-field approximation to estimate equilibrium population density. These methods are able to include features such as unequal dispersal probabilities to different sites in the neighborhood, e.g., as would arise when dispersers have a fixed rate of mortality per distance traveled from the parent site. We also show how urn models allow one to generalize these results to two species competing for space.

Appendix

Let M be the total number of sites available. Let x ₁ and x ₂ be the fraction of occupied sites by species 1 and 2, respectively. Let γ ₁ and γ ₂ be the respective probabilities of survival per unit of time, and ϕ the probability that a site newly occupied by both species simultaneously, will be occupied by species 1, regardless of the amount of each type. Before reproduction, deaths occur and the expected fraction of occupied sites is now x ₁ γ ₁ and x ₂ γ ₂. In the equilibria, the fraction of deaths of each species has to be replaced at the next unit of time by the newly occupied sites of each species. That is, the expected fraction of newly occupied sites at the next unit of time must be equal to x ₁ (1 − γ ₁) and x ₂ (1 − γ ₂) for each species. These newborns can only be produced by those that survived.

Let α ₁ be the fraction of empty sites that will receive a seed from species 1. Let α ₂ be the equivalent for species 2. The fraction of newly occupied sites by species 1 is:

$$ \alpha_1-\alpha_1 \alpha_2+\alpha_1 \alpha_2 \phi = \alpha_1((1-\alpha_2)+\alpha_2\phi) $$

(14)

Now, the probability that a seed will fall in an empty site is θ = 1 − x ₁ γ ₁ − x ₂ γ ₂. Using Eq. 9 with b = Mx ₁ γ ₁ and N = M, we see that the Mx ₁ γ ₁ survivors will produce the same number of seeds that will fall on a fraction

$$ f(x_1,\gamma_1)=1-e^{-x_1 \gamma_1} $$

of the total sites on the grid. Many of those sites are already occupied by survivors of species 1 and 2 in the previous unit of time (and we assume cannot be displaced by the newborns). From this fraction, a fraction θ will fall in an empty site. Thus, we arrive to an expression for α ₁, the fraction of empty sites that will receive a seed from species 1:

$$ \alpha_1 = f(x_1,\gamma_1)\theta $$

(15)

We can obtain α ₂ using a similar argument. Combining this result with Eq. 14, we arrive to the following equalities:

$$ \begin{array}{rll} x_1 (1-\gamma_1)&=&\alpha_1((1-\alpha_2)+\alpha_2\phi)\\ x_2 (1-\gamma_2)&=&\alpha_2((1-\alpha_1)+\alpha_1 (1-\phi)) \end{array} $$

which after simplification yield Eqs. 11 and 12.