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Descripción
A large fraction of living organisms exhibit movement at some stage of their life cycle. Such movement, however, may take different forms. While migration involves periodic displacements with return (Dingle, H., Drake, V.A., BioScience 57, 113–121, 2007), such as the annual routes of birds, dispersal is characterized by the spatial separation of individuals (Begon, M., Townsend, C.R., Harper, J.L., Ecology: From Individuals to Ecosystems, Blackwell, 2006), such as the movement of birds between islands in search of a suitable habitat.
Dispersal is a natural process that has a significant impact on population behavior. It may be influenced by biotic factors, such as interactions with other organisms or chemical cues (chemotaxis) (Painter, K.J., J. Math. Biol. 70, 295–318, 2015; Murray, J.D., Mathematical Biology I, Springer, 2002), as well as abiotic factors, such as interactions with the habitat itself. Understanding this process can help predict population distribution patterns and, consequently, identify mechanisms that facilitate or hinder movement across the environment.
In this work, we propose a mathematical model to describe the dispersal of nonequilibrium metapopulation by incorporating three fundamental mechanisms: (i) spatially heterogeneous diffusion, (ii) nonlocal directed transport modeled through an interaction integral term, and (iii) logistic population growth. The model is formulated as a convection–diffusion–reaction partial differential equation with a nonlocal term, allowing the capture of attraction and repulsion effects among individuals across space.
Let $\rho(x,t)$ denote the population density in a one-dimensional habitat $[0,L]$ over the time interval $[0,T]$. The dispersal dynamics with nonlocal interactions are described by the integro-differential convection–diffusion–reaction equation
\begin{equation}
\partial_t \rho = - \nabla (J_{ran} + J_{int}) + f(\rho),
\end{equation}
where $J_{ran}$ represents the local diffusive flux (random motion), $J_{int}$ the nonlocal interaction flux (directed movement), and $f(\rho)$ the reaction term. The diffusive flux is given by
\begin{equation}
J_{ran} = -D(x)\partial_x \rho,
\end{equation}
with heterogeneous diffusivity
\begin{equation}
D(x) = D_0 e^{-K(x-x_0)^2},
\end{equation}
where $D_0>0$ is a constant, $K$ is associated with the environmental carrying capacity, and $x_0$ denotes the point of highest mobility, reflecting habitat heterogeneity.
The interaction flux is defined as
\begin{equation}
J_{int} = \omega \ \rho \ p(\rho) \int_{\mathbb{R}} \frac{s}{|s|} \ \Omega(|s|,\xi,\mu) \ \rho(x+s,t)\ ds,
\end{equation}
where $\omega$ measures the permeability of the medium, $p(\rho)=1-\rho/P$ is a packing function, and $\Omega$ is the interaction kernel
\begin{equation}
\Omega(|s|,\xi,\mu)=\frac{\mu}{\xi}e^{-|s|/\xi},
\end{equation}
with $\mu$ and $\xi$ representing the strength and range of interactions, respectively.
The reaction term is logistic:
\begin{equation}
f(\rho)=r\rho\left(1-\frac{\rho}{K}\right),
\end{equation}
where $r$ is the intrinsic growth rate.
The model enables the investigation of how heterogeneous diffusion, population growth, and nonlocal interactions influence spatial distribution and pattern formation.
A nondimensionalization procedure simplifies the model, followed by qualitative analysis of equilibrium solutions and stability, focusing on parameters associated with diffusion, carrying capacity, and strength and range of interactions. Finally, the Method of Lines is employed to simulate dispersal scenarios. Results indicate that, depending on strength and range of the interaction force, distinct aggregation patterns may emerge over time, provided these effects overcome the homogenizing tendency of population spreading within the habitat.