Abstract
The interaction between plants and herbivores is one of the most fundamental processes in ecology. Discrete-time models are frequently used for describing the dynamics of plants and herbivores interaction with non-overlapping generations, such that a new generation replaces the old at regular time intervals. Keeping in view the interaction of the apple twig borer and the grape vine, the qualitative behaviour of a discrete-time plant–herbivore model is investigated with weak predator functional response. The topological classification of equilibria is investigated. It is proved that the boundary equilibrium undergoes transcritical bifurcation, whereas unique positive steady-state of discrete-time plant–herbivore model undergoes Neimark–Sacker bifurcation. Numerical simulation is provided to strengthen our theoretical discussion.
Highlights
In mathematical biology, plant–herbivore models are basically modifications of prey– predator models [8]
Kartal [31] investigated the dynamical behaviour of a plant–herbivore model including both differential and difference equations
Direction and existence criteria for Neimark–Sacker bifurcation are investigated at positive steady-state of plant–herbivore model (1)
Summary
Plant–herbivore models are basically modifications of prey– predator models [8]. Kartal [31] investigated the dynamical behaviour of a plant–herbivore model including both differential and difference equations. Li et al [37] discussed period-doubling and Neimark–Sacker bifurcations for a plant–herbivore model incorporating plant toxicity in the functional response of plant–herbivore interactions. Taking into account the interaction of the apple twig borer and the grape vine [3], a mathematical model for plant–herbivore interaction based on weak predator functional. Taking into account strong predator functional response, Din [12] discussed global stability of system (1), and Khan et al [32] investigated Neimark–Sacker bifurcation for system (1) with strong predator functional response.
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