On the stability of fractional order Leslie-Gower type model with non-monotone functional response of intermediate predator

Abstract

In this paper, an attempt is made to understand the dynamics of a fractional order three species Leslie-Gower predator prey food chain model with simplified Holling type IV functional response by considering fractional derivative in Caputo Sense. First, we prove different mathematical results like existence, uniqueness, non-negativity and boundedness of the solutions of fractional order dynamical system. The dissipativeness of the solution of the FDE system is discussed. Further, we investigate the Local stability criteria of all feasible equilibrium points. Global stability of the interior equilibrium point have also been discussed here. Using realistic parameter values, numerically it has been observed that the fractional order system shows more complex dynamics, like chaos as fractional order becomes larger. Analytical results are illustrated with several examples in numerical section.

Figures (8)

• Figure 1: Bifurcation diagram of system (\ref{['Tritophic fractional order model']}) for the $X$ population with respect to $a_0$ in $(0.25, 0.5)$ with different fractional orders $m = 0.95,~ 0.75$ (Fig. 1(b) and 1(c)) and integer order $m = 1$ (Fig. 1(a)). Here $b_0 = 0.075,~ a_1 = 0.105,~ d_1 = d_2 = 10.0,~ d_3 = 20.0,~ v_0 = 1.0,~ v_1 = 2.0,~ v_2 = 0.405,~ v_3 = 1.0$ with $c_3 = 0.047$.
• Figure 2: The trajectory and phase portrait of system (\ref{['Tritophic fractional order model']}) with different fractional orders $m = 0.95,~ 0.75$ (Fig. 2(c) - 2(f)) and integer order $m = 1$ (Fig. 2(a) - 2(b)). We observe that unstable behavior of our system changes to stability with decreasing of fractional order $m$. All the parameters are same as in example 1 with $a_0 = 0.47$ and $c_3 = 0.047$.
• Figure 3: The trajectory and phase portrait of system (\ref{['Tritophic fractional order model']}) with different fractional orders $m = 0.65,~ 0.60 < \frac{2}{3}$ (Fig. 3(a) - 3(d)). We observe that the solution converges to interior equilibrium point for any values of $m < \frac{2}{3}$. It reaches to equilibrium value more slowly as the value of $m$ becomes smaller. All the parameters are same as in example 1 with $a_0 = 0.27$ and $c_3 = 0.047$.
• Figure 4: The trajectory and phase portrait of system (\ref{['Tritophic fractional order model']}) with different fractional orders $m = 0.85,~ 0.75$ (Fig. 4(c) - 4(f)). We observe that the solution converges to interior equilibrium point for any values of $m < \frac{2}{3}$. It reaches to equilibrium value more slowly as the value of $m$ becomes smaller. All the parameters are same as in example 1 with $a_0 = 0.35$ and $c_3 = 0.047$.
• Figure 5: The trajectory and phase portrait of system (\ref{['Tritophic fractional order model']}) on $XY$ plane with different fractional orders $m = 0.95,~ 0.85 > \frac{2}{3}$ (Fig. 5(a) - 5(d)). We observe that the $Y$ population becomes unstable for different values of $m > \frac{2}{3}$. Here $b_0 = 0.03,~ a_1 = 0.001,~ c_3 = 0.047,~ d_1 = d_2 = 10.0,~ d_3 = 20.0,~ v_0 = 0.85,~ v_1 = 2.5,~ v_2 = 2.2,~ v_3 = 1.0$ and initial values are same as in example 1 with $a_0 = 0.15$.