Periodic solutions in a herbivore-plant system with time delay and spatial diffusion

• We presented a mathematical model on the interaction of herbivore and plants.
• Hopf bifurcation is obtained in an herbivore-plant model.
• We show the herbivore outbreak due to time delay.
• We provide a new mechanism on outbreak of herbivore.

Empirical studies indicate that many populations of herbivorous insects exhibit periodic outbreaks, but the intrinsic causes of this behavior are not well understood. Thus, in this study, we investigated a herbivore-plant system with time delay based on reaction-diffusion equations. Using normal formal theory and the center manifold theorem for partial functional differential equations, we show that this model exhibits the property of Hopf bifurcation. Therefore, interactions between the time delay and spatial diffusion will induce periodic outbreaks in herbivore populations. These results may suggest a new mechanism for herbivore outbreaks.