Interpreting how nonlinear diffusion affects the fate of bistable populations using a discrete modelling framework

Abstract

Understanding whether a population will survive or become extinct is a central question in population biology. One way of exploring this question is to study population dynamics using reaction–diffusion equations, where migration is usually represented as a linear diffusion term, and birth–death is represented with a nonlinear source term. While linear diffusion is most commonly employed to study migration, there are several limitations of this approach, such as the inability of linear diffusion-based models to predict a well-defined population front. One way to overcome this is to generalize the constant diffusivity, D , to a nonlinear diffusivity function D ( C ) , where C > 0 is the population density. While the choice of D ( C ) affects long-term survival or extinction of a bistable population, working solely in a continuum framework makes it difficult to understand how the choice of D ( C ) affects survival or extinction. We address this question by working with a discrete simulation model that is easy to interpret. This approach provides clear insight into how the choice of D ( C ) either encourages or suppresses population extinction relative to the classical linear diffusion model.

Curvature dependences of wave propagation in reaction–diffusion models

“…The latter contribution becomes time independent if D = 0 for any choice of the reaction term F(u). These behaviours are confirmed by the explicit long-time solutions of Skellam's model (F(u) = λu), the saturated growth model (F(u) = λ(1 − u)), as well as the strong Allee source term (F(u) = λu(u − a)(1 − u)), which is a common model for bistable populations [4,41,45].…”

“…We also consider a bistable source term Ffalse(ufalse)=λufalse(u−afalse)false(1−ufalse) with 0<a<1, a common model for population growth subject to a strong Allee effect. The rest state u=0 and excited state u=1 in this model are both stable, yet this model still exhibits travelling-wave transitions from u=0 to u=1 or from u=1 to u=0 depending on a and on the shape and size of initial conditions [4,41].…”

“…− a , in one-dimensional infinite space under appropriate boundary and initial conditions [4]. In higher dimensions, the invasion of available space by the solution depends on the size and the shape of the initial condition [4,41]. Numerical simulations of this model are shown in figure 14 with a = 2/5 and λ = 1/(1 − a) = 5/3 so that F(u) ∼ 1 − u as u → 1 and c > 0.…”

Efficient inference and identifiability analysis for differential equation models with random parameters

“…In the supplementary material (fig. S4), we demonstrate a pathological example where our model performs poorly, by approximating the solution to the logistic model with a strong Allee effect [53]. The distribution of the initial condition is chosen so that approximately 16% of model realisations lead to population extinction, whereas 84% lead to logistic growth to carrying capacity.…”

From random walks on networks to nonlinear diffusion

“…More specifically, if there is an agent in node i, we assume that the willingness to leave such node is given by a function f , which depends on the local density, n i /K; and the attractiveness of a neighbouring node j is given by another function g which depends on the density at this node, n j /K. These functions then measure the influence of crowding on motility [37]. Note that this framework is different from other biased random-walks on networks where the bias depends on the network topology rather than on how crowded a particular node is [38][39][40][41].…”