Modelling hiding behaviour in a predator-prey system by both integer order and fractional order derivatives
Section snippets
Ecological background and model formulation
Mathematical modelling is a useful tool to signify the dynamical relationship among the individuals of an ecosystem under consideration. The primary objective of ecosystem modelling is to ensure that the proposed mathematical model can accurately reproduce well-known system behaviour in the system under study. It is worthy to mention here that in the study of predator-prey dynamics mathematically, predation of prey by predators has taken a major part of the long standing research works in the
Essential preliminaries
Before going to model analysis, it is foremost fact to state some essential theories of fractional calculus based on Caputo derivative as our proposed model system is based on Caputo fractional order differential equations. For this purpose, at first, let us state the definition of Caputo derivative as follows
Caputo Derivative
Definition 2.1 (Petráš, 2011) Caputo derivative of a function with order α > 0 in fractional analysis is defined aswhere Γ(*)
Well-posedness
Here, we will establish the well-posedness of system (1.7) both for integer order (α = 1) and fractional order model (0 < α < 1) respectively one by one as follows:
Equilibria and their conditions of existence
Since equilibrium points remain same irrespective of integer or fractional order sense, so, we have obtained all the equilibrium points of model system (1.7) as follows:
- (i)
Trivial equilibrium point E0(0, 0),
- (ii)
Axial equilibrium point E1(k, 0),
- (iii)
Interior equilibrium point E*(x*, y*) where and y* are to be acquired from the underlying equationwhere
Existence of local bifurcations
In this section, we are going to explore the existence of different types of local bifurcations for the model system (1.7) w.r.t. different parameters of our interest under different cases of the system.
Influence of hiding behaviour
In this section, the influence of hiding behaviour to both the prey and predator has been investigated for integer order non-delayed model system (1.7). At first the derivatives with respect to the hide parameter h has been calculated for both prey and predator biomass at interior equilibrium point E*(x*, y*) and we obtain,where
Numerical simulation and discussion
In this section extensive numerical simulation have been carried out using the software Matlab (Kwon and Bang, 2018) and Matcont (Dhooge et al., 2003) to validate all the systematic findings of our model systems. Here, it should be mentioned that, for simulation purpose, we have used different inbuilt tools such as Ode45 which is based on RK4 method for ‘integer order non-delayed system’, dde23 for ‘integer order delayed system’ which is based on explicit ‘Runge-Kutta (2, 3) scheme’, fde12
Conclusion
Refuge of prey to evade risk of predation is a strategic defense mechanism adopted by prey species. But, most of the articles consider that prey usually refuges at a constant rate. Due to practical evidence, it is proved that the amount of prey refuge must be a function of predator's abundance. In this view, a predator-prey model consisting of a single prey and single predator species has been considered incorporating prey refuge. But, rather than considering the prey refuge is constant, we
Declaration of Competing Interest
The authors declare that they have no conflict of interest regarding the publication of this article.
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