Article Contents
Article Contents

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

The work of the third author (Shivam Saxena) is supported by Council of Scientific and Industrial Research (No.09/013(0721)/2017)

• The manuscript aims to investigate the qualitative analysis of a plankton-fish interaction with food limited growth rate of plankton population and non-constant harvesting of fish population. The ecological feasibility of population densities of both plankton and fish in terms of positivity and boundedness of solutions is shown. The conditions for the existence of various equilibrium points and their stability are derived thoroughly. This study mainly focuses on how the harvesting affects equilibrium points, their stability, periodic solutions and bifurcations in the proposed system. It is shown that the system exhibits saddle-node bifurcation in the form of a collision of two interior equilibrium points. Existence conditions for the occurrence of Hopf-bifurcation around interior equilibrium points are discussed. Lyapunov coefficients are examined to check the stability properties of these periodic solutions. We have also plotted the bifurcation diagrams for saddle-node, transcritical and Hopf bifurcations. A detailed algorithm for the occurrence of Bogdanov-Takens bifurcation is derived and finally some numerical simulations are also carried out to validate the theoretical results. This work suggests that the harvesting of fish population can change the dynamics of the system, which may be useful for the ecological management.

Mathematics Subject Classification: Primary: 70K05, 34C23, 34D20, 92D40; Secondary: 37G15.

 Citation:

• 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 $n$

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