In recent decades, ungulates have strikingly increased and expanded their ranges in many parts of the world. Ungulate population dynamics often follows a process known as irruption; initially the population increases gradually and then grows rapidly to peak abundance, and finally crashes to lower abundance. Overabundant deer herds can extirpate preferred plant species and alter natural plant communities into those dominated by less palatable species. Although some mathematical models have been proposed to explain irruptive dynamics of ungulates, no models included dynamics of palatable and unpalatable plants. Thus, we consider a simple three-species Lotka-Volterra model of a palatable and an unpalatable plant species and an herbivore species to understand how the existence of palatable and unpalatable plants affects herbivore dynamics, under the following assumptions; (i) herbivore can grow feeding on abundant palatable plant; (ii) unpalatable species is inferior to the palatable in competition because of the cost of anti-herbivore defense; (iii) However, reduced abundance of the palatable species due to herbivory allows the unpalatable to increase at the equilibrium with the palatable plant and herbivore.
Results/Conclusions
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 state (1) is unstable because the palatable plant invades it. The steady state (2) is unstable because the herbivore invades it. The steady state (3) is unstable because the unpalatable plant invades it. Finally, the equilibrium (4) becomes unstable if the self-regulation of the palatable plant is weak, and population oscillations appear. In these oscillations, the herbivore population increases rapidly if the palatable plant is initially abundant. Then, the herbivore grows and extirpates the palatable plant, and the unpalatable plant dominates the plant community for a long time until the palatable plant recovers from the damage due to overgrazing. This cycle seems to describe the irruptive dynamics of ungulates quite well. Thus, the existence of multiple unstable (not stable) equilibria may be essential to explain the irruptive dynamics of ungulates and the resultant dominance of unpalatable plant species after the crash of ungulate populations. 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.
