Reaction Diffusion Population Model Research Articles

Wavefronts and global stability in a time-delayed population model with stage structure

We formulate and study a one–dimensional single–species diffusive–delay population model. The time delay is the time taken from birth to maturity. Without diffusion, the delay differential model extends the well–known logistic differential equation by allowing delayed constant birth processes and instantaneous quadratically regulated death processes. This delayed model is known to have simple global dynamics similar to that of the logistic equation. Through the use of a sub/supersolution pair method, we show that the diffusive delay model continues to generate simple global dynamics. This has the important biological implication that quadratically regulated death processes dramatically simplify the growth dynamics. We also consider the possibility of travelling wavefront solutions of the scalar equation for the mature population, connecting the zero solution of that equation with the positive steady state. Our main finding here is that our fronts appear to be all monotone, regardless of the size of the delay. This is in sharp contrast to the frequently reported findings that delay causes a loss of monotonicity, with the front developing a prominent hump in some other delay models.

Diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations

For abstract functional differential equations and reaction–diffusion equations with delay, an exponential ordering is introduced which takes into account the spatial diffusion. The induced monotonicity of the solution semiflows is established and applied to describe the threshold dynamics (extinction or persistence/convergence to positive equilibria) for a nonlocal and delayed reaction–diffusion population model.

Persistence, extinction, and critical patch number for island populations.

Sufficient conditions are derived for persistence and extinction of a population inhabiting several islands. Discrete reaction-diffusion population models are analyzed which describe growth and diffusion of a population on a group of islands or a patch environment. A critical patch number is defined as the number of islands below which the population goes extinct on that group of islands. It is shown that population persistence on one island leads to population persistence for the entire archipelago. Both single-species and multi-species models are discussed.

On Volterra’s Population Equation

On Volterra’s Population Equation