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{\bf {Predator-prey system with seasonally varying additional food to predators}}
\vspace{0.6cm} Banshidhar Sahoo
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{\it Department of Mathematics, Daharpur A.P.K.B Vidyabhaban, Paschim Medinipur, West Bengal, India, e.mail:banshivu@gmail.com}\\
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\vspace{0.9cm} {\bf Abstract:}
$~~~$We have proposed a three species food chain model with additional food to predators. We have studied the dynamics of this predator-prey model with seasonally varying additional food to predators. Bifurcation analysis of the proposed model is done with respect to quality and quantity of additional food, amplitude of oscillation and angular frequency of oscillations. The bifurcation analysis of our model revealed the vital role of seasonality parameters in the controllability of the predator-prey system. Our analysis predicts that seasonality parameters are very important to determine the quality and quantity of additional food for controlling the dynamics of a food chain model.\\
{\bf Keywords}:~ Predator-prey; Additional food; Seasonal Variation; Periodic oscillations; Bifurcation; Chaos. \\
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{$1.$ \bf Introduction}\\
$~~~~$ The study of ecological systems subject to seasonal variation is important for both theoretical and experimental ecologists. Most of the models are assumed with constant environment. The constant environment assumption is rarely the case in real life. It is natural to identify the functional role that seasons play on the behavior of population communities and to understand the relationship between the magnitude of the seasonal variations and the complexity of the ecosystem. There were several studies which investigated the interactions between seasonality and internal biological rhythms of simple predator-prey systems [1-3]. These studies show that seasonality has impotant consequences, such as the existence of multiple attractors, catastrophes and chaos. Some studies [4, 5] suggest that chaos should serve as a representation of how real world ecosystems evolve. However, efforts by field ecologists [6] to observe chaos in natural systems have brought negative results. In these studies, the effects of seasonal variation of critical parameters on the dynamical behavior of the systems were investigated. But, the control of chaotic behaviour using seasonal variation in critical parameters of the systems was not examined.
$~~~~~$Hastings and Powell (HP) [7] proposed a chaotic tri-trophic food chain model with holling type-II functional response. After the work of HP [7], many researchers explored their model by including various ecological factors to obtain regular behaviour from the system. In this paper, we introduce a three species predator-prey model with additional (alternative) food to predators. These additional food is assumed to be either non-reproducing prey or some food source. Many experimentalists and theoreticians have investigated the effect of supplying alternative food to predators in a predator-prey system [8-13]. Srinivasu et al. [14], observed that, for a chosen quality and quantity of the additional food the asymptotic state of a solution of the system can either be an equilibrium or a limit cycle. Sahoo [15] applied the concept of additional food on a predator-prey model with different growth rates and different functional response for showing stabilize effects on the system. Recently, Sahoo [16] reported that for biological conservation, additional food plays an important role for servival of consumer species in an ecosystem.
$~~~~~$In this paper, we have studied a predator-prey system with additional food for predators considering seasonal variation of quality of additional food. We have analyzed the behaviour of our proposed model through bifurcation analysis. We have done bifurcation analysis of our model with respect to bifurcation parameters as quality and quantity of additional food, amplitude of oscillations and angular frequency of oscillations of quality of additional food.\\
{\bf $2.$ Model Formulation}\\
$~~~~$ The famous HP [7] model with pairwise interactions between three species, namely, $X$, $Y$, $Z$, which incorporates a Holling type-II functional interactions in both consumer species is the following\\
\begin{eqnarray}
\frac{dX}{dT} & = & R_{0}X(1-\frac{X}{K_0})-C_1A_1\frac{XY}{B_1+X}\nonumber \\
\frac{dY}{dT} & = & A_1\frac{XY}{B_1+X}-A_2\frac{YZ}{B_2+Y}-D_1Y\\
\frac{dZ}{dT} & = & C_2A_2\frac{YZ}{B_2+Y}-D_2Z\nonumber
\end{eqnarray}
$~~~~$Here $X$ are the numbers of species at lowest level of the food chain, $Y$ the size of the species that preys upon $X$ and $Z$ the size of the species that preys upon $Y$. Here $T$ is time. The constant $R_0$ is the \textquotedblleft intrinsic growth rate\textquotedblright and the constant $K_0$ is the \textquotedblleft carrying capacity\textquotedblright of the species $X$. The constant $C_1^{-1}$ and $C_2$ are conversion rates of prey to predators for species $Y$ and $Z$ respectively; $D_1$ and $D_2$ are constant death rates for species $Y$ and $Z$ respectively. The constants $A_i$ and $B_i$ for $i=1, 2$ are maximal predation rate and half saturation constants for $Y$ and $Z$ respectively.\\
$~~~~~$ If $h_1$ and $e_1$, $e_2$ are constants representing handling time of the predators $Y$, $Z$ per prey
item and ability of the predators to detect the prey. Then $A_i$ and $B_i$, represent the maximum predation rate and half saturation values of the predators $Y$, $Z$, to be $1/h_1$ and $1/e_1h_1$, $1/e_2h_1$ respectively.
Hastings and Powell [7] demonstrated that chaos is possible for a simple biologically reasonable, continuous-time, three species food chain model in certain region of parametric space.\\
$~~~~$ Now, we modify the model (1) by introducing \textquotedblleft additional food\textquotedblright to predators population. We make the following assumptions:\\
$(a)$ Predators are provided with additional food of constant biomass $A$ which is distributed uniformly in the habitat.\\
$(b)$ The number of encounters per predator with the additional food is proportional to the density of the additional food.\\
$(c)$ The proportionality constant characterizes the ability of the predator to identify the additional food.\\
$(d)$The handaling time of the both predators per unit quantity of additional food are same.\\
$~~~~$ With the above assumptions, the HP model (1) takes the following form:
\begin{eqnarray}
\frac{dX}{dT} & = & R_{0}X(1-\frac{X}{K_0})-C_1A_1\frac{XY}{B_1+\alpha \mu A+X}\nonumber \\
\frac{dY}{dT} & = & A_1\frac{(X+\mu A)Y}{B_1+\alpha \mu A+X}-A_2\frac{YZ}{B_2+\alpha \nu A+Y}-D_1Y\\
\frac{dZ}{dT} & = & C_2A_2\frac{(Y+\nu A)Z}{B_2+\alpha \nu A+Y}-D_2Z\nonumber
\end{eqnarray}
$~~~~~$ If $h_2$ represents the handaling time of both the predators $Y$, $Z$ per unit quantity of additional food and $e_3$, $e_4$ respectively represent the ability for the predators $Y$, $Z$ to detect the additional food, then we have $\mu=e_3/e_1$, $\nu=e_4/e_2$ and $\alpha=h_2/h_1$. The terms $\mu A$ and $\nu A$ represent effectual additional food for the predators $Y$ and $Z$ respectively.\\
$~~~~~~$ We nondimensionalize the system (2) with
$x=\frac{X}{K_0}$, $y=\frac{Y}{K_0}$ , $z=\frac{Z}{K_0}$ , $t=R_0T$ and obtain the following system \\
\begin{eqnarray}
\frac{dx}{dt} & = & x(1-x)-\frac{a_1x}{1+\alpha \xi +b_1x}y\nonumber \\
\frac{dy}{dt} & = & \frac{\beta(x+c \xi)}{1+\alpha \xi +b_1x}y-\frac{a_2y}{1+\alpha \eta +b_2y}z-d_1y\\
\frac{dz}{dt} & = & \frac{\gamma(y+e \eta)}{1+\alpha \eta+b_2y}z-d_2z\nonumber
\end{eqnarray}
where\\
$a_1=\frac {C_1A_1K_0}{R_0B_1}$, $a_2=\frac {A_2K_0}{B_2R_0}$, $b_1=\frac {K_0}{B_1}$, $b_2=\frac {K_0}{B_2}$, $\beta=\frac {A_1K_0}{B_1R_0}$, $\gamma=\frac{C_2A_2K_0}{R_0B_2}$, $c=\frac {B_1}{K_0}$, $e=\frac{B_2}{K_0}$, $\xi=\frac{\mu A}{B_1}$, $\eta=\frac{\nu A}{B_2}$, $d_1=\frac{D_1}{R_0}$, $d_2=\frac{D_2}{R_0}$.\\\\
$~~~~~~$Here $\alpha$ represents the \textquotedblleft quality\textquotedblright of the additional food ( ratio between predator's handling time towards additional food and prey item) and $\xi$, $\eta$ represent the \textquotedblleft quantity\textquotedblright of the additional food for the intermediate predators and top-predators respectively. The parameters $\alpha$, $\xi$ and $\eta$ are the paramaters which characterize the additional food. \\
$~~~~$Most natural environments are variable, and in response, birth rates, death rates, and other vital parameters vary greatly in time. Therefore, a constant supply of additional food to predators in a system is not relistic in ecology. When the environmental fluctuation is taken into account, a model must be more difficult to analyze in general. So, due to seasonal variation, the supply of additional food to predators must be variable. Here, we assume that the quantity of additional food $\xi$ and $\eta$ are varied periodically for seasonal reason. We use $\xi=\xi_0(1+\delta cos(\omega_1t))$ and $\eta=\eta_0(1+\delta cos(\omega_2t))$, where $\xi_0$ and $\eta_0$ are constants. Here $\delta$ is the amplitude of oscillations and $\omega_1$, $\omega_2$ are the angular frequency of oscillations of $\xi$ and $\eta$ respectively . Therefore, with above assumption, the model (3) becomes\\
\begin{eqnarray}
\frac{dx}{dt} & = & x(1-x)-\frac{a_1x}{1+\alpha \xi_0(1+\delta cos(\omega_1t)) +b_1x}y\nonumber \\
\frac{dy}{dt} & = & \frac{\beta(x+c \xi_0(1+\delta cos(\omega_1t)))}{1+\alpha \xi_0(1+\delta cos(\omega_1t)) +b_1x}y-\frac{a_2y}{1+\alpha \eta_0(1+\delta cos(\omega_2t)) +b_2y}z-d_1y\\
\frac{dz}{dt} & = & \frac{\gamma(y+e \eta_0(1+\delta cos(\omega_2t)))}{1+\alpha \eta_0(1+\delta cos(\omega_2t))+b_2y}z-d_2z\nonumber
\end{eqnarray}
$~~~~$ Now, we shall analyze the dynamics of the system (4) under the variation of quality and quantity of additional food, amplitude and frequency of oscillations of quantity of additional food. The system (4) has to be analyzed with the following initial conditions: $x(0)>0$, $y(0)>0$, $z(0)>0$.\\
{\bf 3. Bifurcation Analysis}\\
$~~~~$We have done bifurcation analysis of the system (4) with the parameter values as $a_1=5.0$, $a_2=0.1$, $b_1=3$, $b_2=2.0$, $c=0.95$, $e=0.85$, $\beta=4.6$, $\gamma=0.08$, $d_1=0.4$, $d_2=0.01$, which remains unchanged throughout simulations. The remaining parameters $\alpha$ (quality of additional food), $\xi$ and $\eta$ (quantity of additional food), $\delta$ (amplitude of oscillations) are varying.\\
{\bf 3.1. Bifurcation Analysis of the Prey Population}\\
$~~~~~$ Bifurcation diagrams of prey population with respect to quality of additional food $\alpha$, quantity of additional food $\xi$, quantity of additional food $\eta$, amplitude of oscillations $\delta$ are shown in figure 1. Figure 1(a) is the bifurcation diagram of the prey population with respect to quality of additional food $\alpha$ taking fixed $\xi=0.05$, $\eta=0.05$, $\delta=0.02$. Figure 1(a) shows that the system has chaotic attractor without additional food $\alpha$. With the increase of quality of additional food the system dynamics becomes periodic and after $\alpha \geq 7.8$, it reaches the steady state. Figure 1(b) is the bifurcation diagram of the prey population with respect to quantity of additional food $\xi$ taking $\alpha=2.0$, $\eta=0.05$, $\delta=0.02$ fixed. Figure 1(b) shows that the system (4) has period-3 behaviour within $0 \leq \xi \leq 0.015$. After $\xi >0.02 $ it has high periodic oscillations and goes to chaotic region after $\xi >0.048$. Bifurcation diagram of the prey population with respect to quantity of additional food $\eta$ taking fixed $\alpha=2$, $\xi=0.05$, $\delta=0.02$ is shown in figure 1(c). From figure 1(c) it is clear that the system has limit cycle oscillations for $0 \leq \eta \leq 0.02$. But, in between $0.02 \leq \eta \leq 0.04$ either periodic oscillations or chaotic bands are occuring. Finaly, the system (4) settles down to period-3 cycle. Figure 1(d) is the bifurcation diagram of the prey population with respect to amplitude of oscillations $\delta$ taking fixed $\alpha=2$, $\xi=0.05$, $\eta=0.05$. Figure 1(d) shows that the system has high periodic or chaotic bands within $0 \leq \delta <1$, but at $\delta=1$ it shows limit cycle oscillation. However, from these bifurcation diagrams, we have observed that proper choice of parameters $\alpha$, $\xi$, $\eta$ and $\delta$ can control the system dynamics.\\
\begin{figure}
\begin{center}
\includegraphics[width=1\textwidth]{season_bifur_prey.eps}
\caption{Bifurcation diagram of prey population with respect to (a) quality of additional food $\alpha \in [0, 8.2]$; (b) quantity of additional food $\xi \in [0, 0.113]$; (c) quantity of additional food $\eta \in [0, 0.18]$; (d) amplitude of oscillations $\delta \in [0, 1]$ of the system (4).}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=1\textwidth]{season_bifur_intpred.eps}
\caption{Bifurcation diagram of intermediate predator with respect to (a) quality of additional food $\alpha \in [0, 8.2]$; (b) quantity of additional food $\xi \in [0, 0.113]$; (c) quantity of additional food $\eta \in [0, 0.18]$; (d) amplitude of oscillations $\delta \in [0, 1]$ of the system (4).}
\end{center}
\end{figure}
{\bf 3.2. Bifurcation Analysis of Intermediate Predator}\\
$~~~~~$ The bifurcation diagrams of the intermediate predator with respect to quality of additional food $\alpha$, quantity of additional food $\xi$, quantity of additional food $\eta$, amplitude of oscillations $\delta$ are shown in figure 2.
Figure 2(a) is the bifurcation diagram of the intermediate predator with resprct to quality of additional food $\alpha$ taking fixed $\xi=0.05$, $\eta=0.05$, $\delta=0.02$. Figure 2(a) shows that the system has chaotic attractor without any quality of additional food $\alpha$. The system shows period-8, period-3, peiod-2, limit cycle oscillations when we increase of additional food $\alpha \geq 2.0$. But, after $\alpha \geq 7.8$ it settles down to steady state. Figure 2(b) is the bifurcation diagram of the intermediate predator with resprct to quantity of additional food $\xi$ taking fixed $\alpha=2$, $\eta=0.05$, $\delta=0.02$. From figure 2(b) we observe that the system (4) has period-3 behaviour within $0 \leq \xi \leq 0.015$. After $\xi >0.02 $ it has high periodic oscillations and goes to chaotic region after $\xi >0.048$. Figure 2(c) is the bifurcation diagram of the intermediate predator with respect to quantity of additional food $\eta$ taking fixed $\alpha=2$, $\xi=0.05$, $\delta=0.02$. The diagram shows that system has limit cycle oscillations within $0 \leq \eta \leq 0.02$. But, in between $0.02 \leq \eta \leq 0.04$ either periodic oscillations or chaotic bands are there. At last it settles down to period-3 oscillation. Figure 2(d) is the bifurcation diagram of the intermediate predator with resprct to amplitude of oscillations $\delta$ taking $\alpha=2$, $\xi=0.05$, $\eta=0.05$ fixed. Figure 2(d) shows that the system has chaotic bands within $0 \leq \delta <1$, but at $\delta=1$ it shows limit cycle oscillation. Therefore from the bifurcation diagrams we conclude that suitable choice of parameters $\alpha$, $\xi$, $\eta$ and $\delta$ are needed to control the dynamics of the system.\\
{\bf 3.3. Bifurcation Analysis of Top-predator}\\
$~~~~~$ The bifurcation diagrams of top-predator with respect to quality of additional food $\alpha$, quantity of additional food $\xi$, quantity of additional food $\eta$, amplitude of oscillations $\delta$ are shown in figure 3. Figure 3(a) is the bifurcation diagram of the top-predator with resprct to quality of additional food $\alpha$ taking fixed $\xi=0.05$, $\eta=0.05$, $\delta=0.02$. Figure 3(a) shows that the system has chaotic attractor without additional food $\alpha$. When we increase of additional food $\alpha$ after $\alpha \geq 3.2$, it has limit cycle oscillation and it settles down to steady state from $\alpha \geq 7.8$. Figure 3(b) is the bifurcation diagram of the top-predator with resprct to quantity of additional food $\xi$ taking fixed $\alpha=2$, $\eta=0.05$, $\delta=0.02$. From figure 3(b) we observe that the system (4) has limit cycle oscillation within $0 \leq \xi \leq 0.015$. After $\xi >0.02 $ it has more periodic oscillations and goes to chaotic region after $\xi >0.048$. Figure 3(c) is the bifurcation diagram of the top-predator with respect to quantity of additional food $\eta$ taking fixed $\alpha=2$, $\xi=0.05$, $\delta=0.02$. The diagram shows that figure 3(c) has limit cycle oscillations within $0 \leq \eta \leq 0.02$. But, in between $0.02 \leq \eta \leq 0.11$ either periodic oscillations or chaotic bands are there. Lastly, it settles down to limit cycle oscillation. Figure 3(d) is the bifurcation diagram of the top-predator with resprct to amplitude of oscillations $\delta$ taking fixed $\alpha=2$, $\xi=0.05$, $\eta=0.05$. Figure 3(d) shows that the system has chaotic bands within $0 \leq \delta <1$, but at $\delta=1$ the system diverges. Therefore we conclude that proper choice of parameters are very important to control the dynamics of the system.\\
\begin{figure}
\begin{center}
\includegraphics[width=1\textwidth]{season_bifur_toppred.eps}
\caption{Bifurcation diagram of top-predator with respect to (a) quality of additional food $\alpha \in [0, 8.2]$; (b) quantity of additional food $\xi \in [0, 0.113]$; (c) quantity of additional food $\eta \in [0, 0.18]$; (d) amplitude of oscillations $\delta \in [0, 1]$ of the system (4).}
\end{center}
\end{figure}
{\bf 3.4. Bifurcation Analysis with respect to Frequency of oscillations}\\
$~~~~~$In this section, we have analysed the bifurcation of prey population, intermediate predator and top-predator with respect to frequency of oscillations $\omega_2$ of the system (4). Here, we take the ratio $\frac {\omega_1}{\omega_2}$ as rational as well as irrational numbers. The figure 4, figure 5, figure 6 are respectively the bifurcation diagram of prey population, intermediate predator and top-predator with respect to $\omega_2 \in [0, 1.5]$ taking the ratio $\frac{\omega_1}{\omega_2}$=Golden number=$\frac{\sqrt{5}+1}{2}$. From the figure 4, figure 5, figure 6 we observe that the system (4) has chaotic attractor within $0 \leq \omega_2 \leq 0.65$. After $\omega_2 > 0.65$, it shows periodic oscillations. Similar behaviour is obtained when the ratio $\frac {\omega_1}{\omega_2}$ is a rational number. Therefore, for high frequency of oscillations of quantity of alternative food the system has periodic behaviour. \\
\begin{figure}
\begin{center}
\includegraphics[width=1.1\textwidth]{season_irr_omega2_prey.eps}
\caption{Bifurcation diagram of prey population with respect to frequency of oscillation $\omega_2 \in [0, 1.5]$, where $\omega_1=\frac{\sqrt{5}+1}{2}\omega_2$ keeping fixed $\alpha=2$, $\xi=0.05$, $\eta=0.05$, $\delta=0.02$ of the system (4) }
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=1.1\textwidth]{season_irr_omega2_intpred.eps}
\caption{Bifurcation diagram of intermediate predator with respect to frequency of oscillation $\omega_2 \in [0, 1.5]$, where $\omega_1=\frac{\sqrt{5}+1}{2}\omega_2$ keeping fixed $\alpha=2$, $\xi=0.05$, $\eta=0.05$, $\delta=0.02$ of the system (4) }
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=1.1\textwidth]{season_irr_omega2_toppred.eps}
\caption{Bifurcation diagram of top-predator with respect to frequency of oscillation $\omega_2 \in [0, 1.5]$, where $\omega_1=\frac{\sqrt{5}+1}{2}\omega_2$ keeping fixed $\alpha=2$, $\xi=0.05$, $\eta=0.05$, $\delta=0.02$ of the system (4) }
\end{center}
\end{figure}
\newpage
{\bf 4. \bf Conclusions}\\
$~~~~~$ In this paper, we have proposed a three species predator-prey model with additional food to predators. We assume the periodic variation of quantity of additional food. The bifurcation analysis of proposed model is done with respect to quality of additional food $\alpha$, quantity of additional food $\xi$ and $\eta$, amplitude of oscillations $\delta$ and angular frequency of oscillations separately. From the bifurcation diagrams, we have observed that for suitable choice of parameters the system dynamics can be controlled. Our analysis confirms that if quantity of additional food is season dependent, then system dynamics may not be controlled through variation of quantity and quality of additional food. We can reach our target state from the system with periodically varying quantity of additional food only if we have knowledge of frequency as well as amplitude of oscillation of the quality of additional food. Therefore we conclude that together with quality and quantity of additional food, the frequency and amplitude of oscillation of quantity of additional food are key finding parameters to control the dynamics of a three species predator-prey system. Therefore for pest managment and biological conservation consideration of seasonal variation of quantity of additional food is urgently required.\\
{\bf 5. \bf References}\\
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\item Sabelis MW, van Rijn PCJ, "When does alternative food promote biological pest control?" In: Hoddle MS (ed) Proc. second int. symp. biol. control of arthropods. 2: 428-437 (2005).
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\item Srinivasu PDN, Prasad BSRV and Venkatesulu M, "Biological control through provision of additional food to predators:a theoretical study", Theor Popul Biol. 72: 111-120 (2007).
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\end{enumerate}
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