The qualitative behavior of a plankton-fish interaction model with food limited growth rate and non-constant fish harvesting

Figure 1.  Zero-growth isoclines along with the interior equilibrium states of the system (6) are depicted in this figure. Here, red and green color curves represent the plankton and fish nullclines respectively. For the values of the parameters $ r = 0.1 $, $ k = 1 $, $ a = 0.3 $, $ m = 0.03 $, $ d = 0.07 $, $ n = 0.23 $, $ c = 0.4 $, we get two equilibrium (A), then the two unique equilibrium collide to reach unique equilibrium (B) and then no equilibrium (C), for $ h = 0.09, 0.1116729526 $ and $ 0.13 $ respectively. If we take $ r = 0.25 $, $ k = 10 $, $ a = 0.2 $, $ m = 0.5 $, $ d = 0.9 $, $ h = 0.2 $, $ c = 0.2 $, $ n = 0.2 $, then the system has unique interior equilibrium point (D)

Figure 2.  In (A) $ L_{2*} $ is stable for $ r = 0.1 $, $ k = 1 $, $ a = 0.3 $, $ m = 0.03 $, $ d = 0.07 $, $ h = 0.11 $, $ c = 0.4 $ and $ n = 0.25 $, if we increase the bifurcation parameter $ n $ to $ n = n^{[H]} = 0.3068476027 $, the system possesses a periodic solution about $ L_{2*} $ which is represented in (B) and finally for $ n = 0.307 $ and keeping rest of the parameters same, the periodic solution collide with the saddle point $ L_{1*} $ to give a homoclinic orbit about $ L_{2*} $ which is shown in (C) where as $ L_{1*} $ remains saddle in each case. Figure (D) represents the bifurcation diagram for $ r = 0.1 $, $ k = 1 $, $ a = 0.3 $, $ m = 0.03 $, $ d = 0.07 $, $ h = 0.11 $ and $ c = 0.4 $ with respect to the bifurcation parameter $ n $

Figure 3.  (A) Bifurcation diagram w.r.t. parameter $ a $ in case of two interior equilibrium. (B) Bifurcation diagram w.r.t. parameter $ a $ in case of one interior equilibrium

Figure 4.  (A) For $ r = 0.1 $, $ k = 1 $, $ a = 0.3 $, $ m = 0.03 $, $ n = 0.23 $, $ d = 0.07 $, $ c = 0.4 $ and $ h = 0.09 $ there are two interior equilibrium points of the system (6). (B) The model (6) attains a unique instantaneous equilibrium point for $ h = h^{[SN]} = 0.1116729526 $ and keeping rest of the values same. (C) For $ h = 0.13 $ and keeping other parameter fixed the system (6) has no equilibrium point. (D) Represents bifurcation curves in $ \lambda_1 $$ \lambda_2 $-plane

Figure 6.  (A) The components of interior equilibrium are plotted to show their stability. The red curves stand for stable branch and green curves stand for unstable branch. (B) This figure depicts the P-components of $ L_1 $ and $ L_{2*} $ for $ r = 2.0 $, $ k = 22 $, $ a = 0.01 $, $ m = 0.02 $, $ n = 0.1 $, $ d = 0.08 $, $ h = 0.05 $, $ c = 0.2 $ as d varies. When $ d<1.51 $ the equilibrium $ L_1 $ is stable while the interior equilibrium $ L_{2*} $ is unstable and when $ d>1.51 $, the interior equilibrium $ L_{2*} $ is stable while the equilibrium $ L_1 $ is unstable

Figure 5.  (A) For $ r = 0.1 $, $ k = 1 $, $ a = 0.3 $, $ m = 0.03 $, $ d = 0.07 $, $ c = 0.4 $, $ h = 0.1116729526 $, $ n = 0.2313139014 $ and $ h = 0.13 $ the system attains an unique instantaneous equilibrium point which is a cusp of co-dimension 2. (B) Here, the trace and determinant of the variational matrix at $ (N_{2*},P_{2*}) $ are plotted in green and red color where their intersection is $ (n_0, h_0) = (0.2313139014, 0.1116729526) $

Figure 7.  In (A) there is a stable point for $ r = 0.25 $, $ k = 10 $, $ a = 0.2 $, $ m = 0.5 $, $ d = 0.9 $, $ h = 0.2 $, $ c = 0.2 $ and $ n = 0.194 $, if we increase the bifurcation parameter $ n $ to $ n = n^{[H]} = 0.1952313043 $, the system possesses a periodic solution which is shown in figure (B) and finally for $ n = 0.23 $ and keeping rest of the parameters same, the periodic solution collide with the saddle point to give a homoclinic orbit which is given in figure (C) and $ L_{1*} $ remains saddle in each case. (D) represents the bifurcation diagram for $ r = 0.25 $, $ k = 10 $, $ a = 0.2 $, $ m = 0.5 $, $ d = 0.9 $, $ h = 0.2 $ and $ c = 0.2 $ with respect to the bifurcation parameter $ n $