| Abstract | In recent decades, ungulate populations have strikingly increased and expanded their ranges in many countries. Ungulate population dynamics often follows a rapid increase to peak abundance and a subsequent crash to much lower abundance, a process known as irruption. Overabundant ungulate herds can extirpate palatable plant species and drive the natural plant communities into those dominated by unpalatable species. Although some mathematical models have explained ungulate dynamics by a limit cycle or a transition between multiple stable states, no models of ungulate outbreaks have included dynamics of palatable and unpalatable plants. Here, we consider a simple three-species Lotka-Volterra model of a palatable and an unpalatable plant species and an herbivore species to understand how competition between palatable and unpalatable plants affects herbivore dynamics. We assume that unpalatable plant species is inferior to the palatable in competition because of the cost of anti-herbivore defense. However, herbivores may reduce the abundance of palatable species and allow the unpalatable species to increase. Under the above assumptions, there exist steady states, at which (1) only the unpalatable plant survives, (2) only the palatable plant survives, (3) the palatable plant and herbivore coexist, and (4) three species of the palatable and unpalatable plants and herbivore coexist. The steady states (1) to (3) are always unstable and the steady state (4) becomes unstable if intraspecific competition of the palatable plant is weak. Then, all of the four steady states are unstable, and population oscillations occur. In these oscillations, the herbivore population increases rapidly when the palatable plant is initially abundant. Then, the herbivore extirpates the palatable plant and crashes. Then, unpalatable plant invades. The unpalatable plant dominates the plant community for long time until the palatable plant recovers from overgrazing. The cycles staying around some critical points for long time resemble homoclinic-like or heteroclinic-like cycles. These cycles seem to describe the irruptive dynamics of ungulates quite well. Thus, the existence of multiple unstable (not stable) steady states may be essential to explain the irruptive dynamics of ungulates and the resultant dominance of unpalatable plant species after the crash of the ungulate population. We will also consider a model that incorporates a carnivore species and investigate effects of the trophic cascade on the dynamics of herbivores and plants. |
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