A MODIFIED LESLIE-GOWER FRACTIONAL ORDER PREY-PREDATOR INTERACTION MODEL INCORPORATING THE EFFECT OF FEAR ON PREY

Presentation of different dynamical behaviour of the model system for $ \alpha=1 $ by considering different value of fear level $ k $. The system dynamics exhibits stable scenario for $ k=0.01 $ which has been plotted by blue lines; for $ k=0.3 $ it shows unstable scenario through periodic oscillation (see red curve) while for $ k=0.5 $ it illustrates stable scenario by washing out the periodic oscillations.

Occurrence of multiple Hopf-bifurcation w.r.t fear level $ k $ in $ [0, 1] $ for $ \alpha=1 $. The first Hopf point occurs for $ k=k^*=0.069 $ while the second Hopf point occurs for $ k=k^*=0.458709 $.

Plot of bifurcation diagram w.r.t $ k $ in $ [0, 1] $ for $ \alpha=1 $. The model system shows stable scenario for $ k<k^*=0.069 $, but for $ k \in [0.069, \; 0.458] $ it shows oscillatory scenario. Interestingly for further increase in fear level $ k $ the system becomes stable.

Relation between period of the bifurcating limit cycle with fear level $ k $ for $ \alpha=1 $. It is noticed that period of the bifurcating limit cycle increases with the increase of fear level.

Exhibits stability of limit cycle oscillation for different values of $ k $ based on the properties of Floquet multipliers for $ \alpha=1 $. Here, modulus of one Floquet multiplier is one and other one is less than one and hence the limit cycle oscillation is stable.

Transcritical bifurcation diagram w.r.t $ k $ for the integer order model system.

Phase portrait diagram under different values of $ \alpha $ for predator induced fear $ k=0.3 $. It is examined that reduction in $ \alpha $ makes the model system stable.

Bifurcation diagram w.r.t $ \alpha $ by taking $ k=0.3 $ and remaining other parameters same as above. It is examined that for the threshold value of fractional order $ \alpha>\alpha^*=0.945 $ the model system exhibits unstable scenario by making sufficiently large difference between maximum and minimum of both the population density while slight decrements in $ \alpha $ makes the system dynamics stable by making maximum and minimum of both the population density same.

Plot of change in mean population biomass of both the species under the variation of model parameter $ \alpha $ considering $ k=0.3 $. It is observed that both prey and predator population start to oscillate when $ \alpha $ just crosses the threshold parametric limit $ \alpha=\alpha^*=0.945 $. For $ 0 <\alpha \leq 0.4 $ mean density of prey species persistently increases and then in the range $ 0.4 <\alpha \leq 0.65 $ it decreases while predator population biomass continuously increases in the range $ 0 <\alpha \leq 0.55 $.

Phase portraits plot of the system for $ \alpha=1 $. Fig.A exhibits the limit cycle oscillation for $ f=0.04 $; Fig.B displays the disappearance of limit cycle oscillation for $ f=0.07 $

Occurrence of multiple Hopf-bifurcation w.r.t $ f $ for the integer order model system.

Bifurcation diagram w.r.t $ f $ for the integer order model system.

Stability of limit cycle oscillation w.r.t $ f $ for the integer order model system.

Phase portrait diagram under different values of $ \alpha $ for $ f=0.04 $ keeping other parameter same as above. We have examined that reduction in $ \alpha $ makes the model system stable.

Bifurcation diagram w.r.t $ \alpha $ for $ f=0.04 $ in $ \alpha \in [0, 1] $. It reveals the fact that for $ \alpha \in [0.95, 1] $ the model system exhibits unstable scenario by producing periodic oscillation of both the population density (existence of maximum and minimum population density), however, slight reduction in $ \alpha $ makes the system free from periodic oscillation and hence displays stable scenario.

Plot of change in mean population biomass of both the species under the variation of model parameter $ \alpha $ for $ f=0.04 $. It is observed that both prey and predator population start to oscillate when $ \alpha $ just crosses the threshold parametric value $ \alpha=\alpha^*=0.95 $. For $ 0 <\alpha \leq 0.4 $ mean density of prey species persistently increases and then in the range $ 0.4 <\alpha \leq 0.65 $ it decreases while predator population biomass continuously increases in the range $ 0 <\alpha \leq 0.55 $.

Phase portraits plot of the system for $ \alpha=1 $. The system dynamics exhibits stable scenario for $ b=0.5 $ as shown in Fig.a; but for $ b=0.9 $ it shows unstable scenario by producing periodic oscillation as displayed in Fig.b.

Bifurcation diagram w.r.t $ b $ in $ [0, 1] $ for the integer order model system. It is examined that the model system shows locally asymptotically stable scenario until $ b $ exceeds the threshold value $ b=b^*=0.707356 $; whenever $ b $ just crosses this threshold parametric limit $ b=b^*=0.707356 $ then the system dynamics changes its qualitative scenario from stable to unstable and it continues for further increase in $ b $.

Stability of limit cycle oscillation through the plot of Floquet multipliers for the integer order counterpart system. Here, it is observed that out of two Floquet multipliers one is always $ 1 $ and the other one is less than $ 1 $ which consequently reveals that the limit cycle oscillations are stable.

Saddle-node bifurcation for $ b=2.13 $ for the integer order counterpart system.

Phase portrait diagram under different values of $ \alpha $ for $ b=1.5 $ keeping other parameters same as above. It is examined that reduction in $ \alpha $ makes the model system stable for $ b=1.5 $.

Bifurcation diagram w.r.t $ \alpha $ for $ b=1.5 $ in $ \alpha \in [0, 1] $. It reveals the fact that for $ \alpha \in [0.86, 1] $ the model system exhibits unstable scenario by producing periodic oscillation of both the population density (existence of maximum and minimum population density), however, slight reduction in $ \alpha $ makes the system free from periodic oscillation and hence displays stable scenario.

Plot of change in mean population biomass of both the species under the variation of model parameter $ \alpha $ considering $ b=1.5 $. It is observed that both prey and predator population start to oscillate when $ \alpha $ just crosses the threshold parametric limit $ \alpha=\alpha^*=0.86 $. For $ 0 <\alpha \leq 0.4 $ mean density of prey species persistently increases and then in the range $ 0.4 <\alpha \leq 0.6 $ it decreases while predator population biomass continuously increases in the range $ 0 <\alpha \leq 0.55 $.