Persistence under Relaxed Point-Dissipativity (with Application to an Endemic Model)

An approach to persistence theory is presented which focuses on the concept of uniform weak persistence. By using the most elementary dynamical systems concepts only, it can be shown that uniform weak persistence implies uniform strong persistence. This even holds under relaxed point dissipativity. Uniform weak persistence can be proved by the method of fluctuation or by analyzing the boundary flow for acyclicity with point dissipativity being only required in a neighborhood of the boundary. The approach is illustrated for a model describing the spread of a fatal infectious disease in a population that would grow exponentially without the disease. Sharp conditions are derived for both host and disease persistence and for host limitation by the disease.