Abstract
We present a simple mathematical model for the dynamics of a successional pioneer–climax system using difference equations. Each population is subject to inter- and intraspecific competition; population growth is dependent on the combined densities of both species. Nine different geometric cases, corresponding to different orientations of the zero-growth isoclines, are possible for this system. We fully characterize the long-term dynamics of the model for each of the nine cases, uncovering diverse sets of potential behaviors. Competitive exclusion of the pioneer species and of the climax species are both possible depending on the relative strength of competition. Stable coexistence of both species may also occur; in two cases, a coexistence state is destabilized through a Neimark–Sacker bifurcation, and an attracting invariant circle is born. The invariant circle eventually disappears into thin air in a heteroclinic or homoclinic bifurcation, leading to the sudden transition of the system to an exclusion state. Neither global bifurcation has been observed in a discrete-time pioneer–climax model before. The homoclinic bifurcation is novel to all pioneer–climax models. We conclude by discussing the ecological implications of our results.
Highlights
Newly established habitat is initially composed of pioneer plants, hardy species that do best at low population densities and that can colonize unpopulated environments (Dalling 2019; Harper 1977; Turner 2001)
We introduced a simple discrete-time pioneer–climax model where the species’ interaction dynamics are built upon principles of ecological succession
We characterized the long-term dynamics of the model in the first in-depth analytical stability analysis of a discrete-time pioneer–climax system, and the stability results were supported by numerical simulations
Summary
Newly established habitat is initially composed of pioneer plants, hardy species that do best at low population densities and that can colonize unpopulated environments (Dalling 2019; Harper 1977; Turner 2001). Climax plants typically cannot persist on their own, and pioneer species act as nurse plants in a commensal interaction (Harper 1977; Selgrade and Namkoong 1990). As population density of the climax species increases, interaction dynamics become more competitive and the climax species tends to thrive (Pandolfi 2019; Selgrade and Namkoong 1990). White bursage and creosote bush from the Sonoran Desert, a pioneer species and a climax species, are one such pair (McAuliffe 1988)
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