Spatiotemporal mutualistic model of mistletoes and birds

A mathematical model which incorporates the spatial dispersal and interaction dynamics of mistletoes and birds is derived and studied to gain insights of the spatial heterogeneity in abundance of mistletoes. Fickian diffusion and chemotaxis are used to model the random movement of birds and the aggregation of birds due to the attraction of mistletoes, respectively. The spread of mistletoes by birds is expressed by a dispersal operator, which is typically a convolution integral with a dispersal kernel. Two different types of kernel functions are used to study the model, one is a Dirac delta function which reflects the special case that the spread behavior is local, and the other one is a general non-negative symmetric function which describes the nonlocal spread of mistletoes. When the kernel function is taken as the Dirac delta function, the threshold condition for the existence of mistletoes is given and explored in terms of parameters. For the general non-negative symmetric kernel case, we prove the existence and stability of spatially nonhomogeneous equilibria. Numerical simulations are conducted by taking specific forms of kernel functions. Our study shows that the spatial heterogeneous patterns of mistletoes are related to the specific dispersal pattern of birds which carry mistletoe seeds.