Qualitative Analysis of Integral Dynamics Model of Age-Structured One-Specious Population with Intra-Species Competition
by
Natalia Hritonenko
Department of Mathematics Prairie View A&M University P.O. Box 4189, Prairie View TX USA 77446

The qualitative analysis of nonlinear Volterra integral equations arising in ecological modeling is provided. A nonlinear integral model for a single-species population is proposed and studied. It takes into account two key intrinsic endogenous factors that influence the population dynamics: age distribution and intra-species competition. The fertility and mortality of individuals depend on their age, so the rational exploitation of populations requires considering their age structure. The intra-species competition reflects an increase in the individual mortality caused by other individuals in their struggle for common external resources and brings a non-linear self-regulatory feedback to the model.

Provided qualitative analysis shows that the solution to the integral model has different behavior depending on the value of a bifurcation parameter. It has trivial and nontrivial positive stationary states. Stability analysis shows that the trivial stationary state is stable initially and becomes unstable when a non-trivial positive stationary state appears. Then the non-trivial positive stationary state changes its stability as the bifurcation parameter grows. The obtained results are discussed and compared with differential and difference models that describe single-species population under similar assumptions. Some initial analysis shows the presence of new bifurcations and oscillatory regimes in the nonlinear integral model (for larger values of the bifurcation parameter. For a specific choice of model functions the integral model is reduced to difference population models of the first and second order that possess oscillatory and chaotic behavior.

Date received: January 31, 2003