Cross-Diffusive Instabilities and Aggregation in Partial Differential Equation Models of Interacting Populations
Open Access
Author:
De Forest, Russ
Graduate Program:
Mathematics
Degree:
Doctor of Philosophy
Document Type:
Dissertation
Date of Defense:
January 21, 2019
Committee Members:
Andrew Leonard Belmonte, Dissertation Advisor/Co-Advisor Andrew Leonard Belmonte, Committee Chair/Co-Chair Timothy Charles Reluga, Committee Member Helge Kristian Jenssen, Committee Member Christopher H Griffin, Outside Member
Keywords:
cross-diffusion normally parabolic systems quasilinear PDE public goods games predator-prey
Abstract:
We propose several systems of quasilinear partial differential equations as spatial models of
interacting biological populations. The key distinctive feature
present in
each model
is a negative self-diffusivity in one of
the populations. Despite the presence of negative self-diffusion,
under our assumptions the resulting models correspond to
normally parabolic systems or degenerate limiting cases of
normally parabolic systems.
We consider a spatial predator-prey
model and show the
existence of a cross-diffusive instability leading to spatial
patterning. Numerical examples are given in one and two dimensions. Our model demonstrates a mechanism by which
prey aggregate in response to predators, potentially reducing their
individual risk of predation.
We also consider several specific spatial models
of polymorphic populations with both a cooperative and exploitative
type in a nonlinear public goods game. Each phenotype is represented by a density and the fitness of each type
depends locally on the density of all types. We demonstrate conditions for
the existence of a cross-diffusive instability, leading to pattern formation
and the advantageous aggregation of the cooperative type.