Permanence in seasonal ecological models with spatial heterogeneity

Date of Award




Degree Name

Doctor of Philosophy (Ph.D.)

First Committee Member

Robert Stephen Cantrell, Committee Chair


Reaction-Diffusion systems are frequently used to model the population dynamics of interacting biological species in bounded spatial regions. The local population growth law for the density of each species in such a system depends in general on time, spatial location and the density of each species. This work examines the question of longterm coexistence in such models in the case in which the growth laws are periodic in time. The criterion for longterm coexistence we employ is the existence of positive (in a sense appropriate to the boundary conditions we impose on the system) asymptotic upper and lower bounds on each component of the system, a criterion we call permanence which is related to a dynamical systems concept called "Abstract Permanence". In this work we establish sufficient conditions for permanence phenomena in several cases. Our approach is to reformulate the system of differential equations as a semidynamical system, apply machinery available in this context to obtain abstract permanence and then to show that abstract permanence implies the aforementioned upper and lower asymptotic bounds on the components of the system. The conditions that we ultimately derive for permanence are expressed in Quantifiable ways in terms of the spectra of linear differential operators associated with the original reaction-diffusion system. In so doing, we connect asymptotic coexistence in such a system to the underlying biological assumptions about the model which are expressed in the parameters and coefficients of these operators. We illustrate our results via two species predator-prey models and three-species competition models. We also show that permanence implies the existence of a componentwise positive periodic orbit (which is not necessary stable).


Biology, Ecology; Mathematics

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