Abstract
ABSTRACTIn many existing predator–prey or plant–herbivore models, the numerical response is assumed to be proportional to the functional response. In this paper, without such an assumption, we consider a diffusive plant–herbivore system with Neumann boundary conditions. Besides stability of spatially homogeneous steady states, we also derive conditions for the occurrence of Hopf bifurcation and steady-state bifurcation and provide geometrical methods to locate the bifurcation values. We numerically explore the complex transient spatio-temporal behaviours induced by these bifurcations. A large variety of different types of transient behaviours including oscillations in one or both of space and time are observed.
Highlights
Plant–herbivore interactions have been extensively studied by many researchers
We study the dynamics of plant–herbivore interactions with plant defense by assuming the numerical response function is not necessarily proportional to the functional response
By adjusting the parameter values, we can have four possible scenarios: (i) neither Hopf bifurcation values σnH nor steady-state bifurcation values σnS exist for positive wave numbers n; (ii) there are Hopf bifurcation values but no steady-state bifurcation values; (iii) both Hopf bifurcation values σnH and steady-state bifurcation values σnS exist and (iv) only steady-state bifurcation values exist and no Hopf bifurcation value exist for any positive wave number n
Summary
Plant–herbivore interactions have been extensively studied by many researchers. Generally speaking, plant–herbivore interactions fall into the category of predator–prey systems. Rich dynamics are possible for predator–prey models with non-monotonic response functions ( called Holling type IV) [21,31], age-structure [5], delays [20,28] and diffusions [7]. Many existing predator–prey models assume that the numerical response is proportional to the functional response, i.e. g = f , where is the conversion coefficient denoting the energetic efficiency in converting consumption into reproduction. We study the dynamics of plant–herbivore interactions with plant defense by assuming the numerical response function is not necessarily proportional to the functional response. If β = σ , (2) reduces to the diffusive predator–prey model in which numerical response is assumed to be proportional to functional response, which has been extensively studied in the literature.
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