Ching, A; Baigent, S; (2019) Manifolds of balance in planar ecological systems. Applied Mathematics and Computation , 358 pp. 204-215. 10.1016/j.amc.2019.04.047.

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Abstract

In the classic 2-species Lotka–Volterra competition model, and more general competitive planar Kolmogorov models, there is a continuous curve called the carrying simplex that links all non-zero steady states and attracts all non-zero population densities. This curve is where the opposing processes of population growth and decline balance. In this paper, we use stability analysis and index theory to show that such a curve also exists when the interactions between two species are more general, such as co-operative or predator-prey, provided that reasonable biologically motivated conditions hold. For example, both species experience intraspecific competition and all population densities remain bounded for all time. We consider systems where there is at most one co-existence steady state. The ‘balance manifold’ is formed of heteroclinic orbits and attracts all non-zero population densities, but unlike its competitive analogue, the curve is no longer necessarily continuously differentiable.

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