1 Introduction

Representing dynamics of population through mathematical models seem to have been initiated by Malthus in early nineteenth century through the development of Malthusian’s model. This model was subsequently modified by introducing linear per-capita growth rate for the population which resulted in the well known logistic model. This modification allowed the population to evolve in such a way that it tends asymptotically to a constant termed as carrying capacity of the environment for the population [1]. In the early twentieth century, Lotka and Volterra independently developed models representing predator—prey interactions. Subsequently, this Lotka–Volterra model has been subjected to several modifications to incorporate the complex relationships that exist between the interacting species. For example, a variety of species dependent functional responses have been developed that enabled realistic representation of the interactions between prey and predator species [2–5]. Similarly, growth rate of the prey species has also been subjected to modifications through incorporation of various Allee effects such as additive Allee effect [6] and multiplicative Allee effect [7]. Although these attempts bring in complexities into the system dynamics, they draw the models closer to reality.

Allee effect is a realistic phenomenon that finds its presence in many fields associated with evolutionary biology. This concept has been formalised and initiated by the work of W.C. Allee in the early twentieth century [8]. Essentially, Allee effect refers to reduction in individual fitness at low population size or density [9]. This effect plays very important role in population dynamics. There are several mechanisms which generate Allee effect in population growth. A classification of various Allee effects characterised by the nature of involved mechanisms is presented in [10]. There are some real world examples [9, 11–16] where the presence of Allee effect is observed. Consequently, analysis of systems involving Allee effect has gained lot of importance in problems associated with various fields such as conservation biology [17, 18], sustainable harvesting [19], pest control, biological control [20], population management [10], biological invasions [21–25], metapopulation dynamics [26, 27], interacting species [28–32]. The review article [33] presents a classification of single species models that are subjected to various Allee effects. Elaborate discussion on Allee effect can be found in the recent article [8]. Although there are many classifications within Allee effect such as strong Allee effect, weak Allee effect, demographic Allee effect etc., dynamics of population in presence of strong and weak Allee effects have been subjected to lot of investigations both mathematically [7, 34–40] and ecologically [8, 41].

Another important concept which is being investigated in the recent past is the dynamics of predator-prey systems when the predator is provided with additional food [42–45]. This concept deserves attention as many predators are more often found to be generalists rather than specialists. Moreover, provision of additional food to one of the interacting species has become an eco-friendly practice in the fields such as biological conservation, bio-remediation, resource management, biological control, pest management etc. Thus studying predator-prey dynamics when the predator is provided with additional food, following consequent changes in qualitative behaviour of the system and its dependance on the characteristics of the additional food such as quality and quantity would not only be mathematically interesting but also of practical importance.

In this article, we consider a predator-prey system where the growth of the prey species is influenced by Allee effect and the predator population is provided with additional food. The considered model can be made to exhibit both strong as well as weak Allee effect with appropriate choice on Allee effect parameter value. Our interest is to understand the role of additional food on the predator-prey dynamics. Comparing the considered system dynamics with the one without additional food reveals several interesting consequences which can be utilised for the purpose of biological control and species conservation.

This article is organised as follows. Section 2 introduces the model and presents a simplified version through the process of non-dimensionalisation. Section 3 presents equilibrium analysis and occurrence of local bifurcations. Global dynamics of the system are demonstrated through bifurcation diagrams and phase portraits in Sect. 4. This is followed by discussion section. The analysis pertaining to various local bifurcations is presented in the “Appendix”.

2 Model

In this paper we consider the following nonlinear system which represents predator-prey interactions

$$\begin{aligned}&\frac{dN}{dT} = rN(N-L)\left( 1 - \frac{N}{K}\right) - g(N,P,A)P, \end{aligned}$$
(1)
$$\begin{aligned}&\frac{dP}{dT} = s(g(N,P,A) + h(N,P,A))P - mP, \end{aligned}$$
(2)
$$\begin{aligned}&P(0), N(0) > 0 \end{aligned}$$
(3)

wherein the growth of the prey is influenced by Allee effect (strong/ weak) and the predator is provided with additional food. Here N(t), P(t) respectively represent prey and predator population densities at time t. K and L (with \(-K < L < K\)) are the carrying capacity and Allee effect threshold of the prey population respectively. r is the intrinsic growth rate of the prey. The parameters s and m, respectively, stand for growth and death rates of the predator. A is the amount of additional food [42], g(N, P, A), h(N, P, A) are the functional responses of the predator towards the prey and additional food respectively. Here, the forms for g(N, P, A), h(N, P, A) are assumed to be

$$\begin{aligned} g(N,P,A)&= \frac{cN}{a+ \alpha \eta A + N}, \end{aligned}$$
(4)
$$\begin{aligned} h(N,P,A)&= \frac{c\eta A}{a+ \alpha \eta A + N} \end{aligned}$$
(5)

where c is the maximum rate of predation, a is the half saturation level of the functional response in the absence of additional food. \(\eta \) represents ability of the predator to detect the prey relative to predator and the term \(\eta A\) stands for quantity of additional food perceptible by the predator relative to prey (effectual additional food level) and \(\alpha \) represents the quality of additional food relative to prey [42]. In the absence of additional food, the functional response g(N, P, A) reduces to the Holling type II functional response and h(N, P, A) reduces to zero. With the functions (4), (5) and denoting \(b=cs\), the model (1)–(2) takes the form

$$\begin{aligned} \frac{dN}{dT}&= rN(N-L)\left( 1 - \frac{N}{K}\right) - \frac{cNP}{a+ \alpha \eta A + N}, \end{aligned}$$
(6)
$$\begin{aligned} \frac{dP}{dT}&= \frac{b(N + \eta A)P}{a+ \alpha \eta A + N} - mP. \end{aligned}$$
(7)

To reduce the number of parameters, we introduce three dimensionless variables x, y and t, and five dimensionless parameters \(\theta , \gamma ,\xi ,\beta \) and \(\delta \) given by \(x = \frac{N}{a}\), \(y = \frac{cP}{ra^2}\), \(t = arT\), \(\theta = \frac{L}{a}\), \(\gamma = \frac{K}{a}\), \(\xi = \frac{\eta A}{a}\), \(\beta = \frac{b}{ra}\) and \(\delta = \frac{m}{ra}\), thereby reduce the system (6)–(7) to the form

$$\begin{aligned} \dot{x}&= x (x - \theta )\left( 1 - \frac{x}{\gamma }\right) - \frac{xy}{1+ \alpha \xi + x}, \end{aligned}$$
(8)
$$\begin{aligned} \dot{y}&= \frac{\beta (x + \xi ) y}{1+ \alpha \xi + x} - \delta y, \end{aligned}$$
(9)
$$\begin{aligned}&x(0) \equiv x_0 \ge 0, y(0) \equiv y_0 \ge 0 \end{aligned}$$
(10)

All the parameters involved with the model are positive except \(\theta \). It is well known that the Allee effect is classified as strong Allee effect and weak Allee effect whenever the parameter \(\theta \) satisfies \(0 < \theta < \gamma \) and \(\theta < 0\) respectively [40].

Defining

$$\begin{aligned} f(x)= & {} \frac{x}{1 + \alpha \xi + x},\\ g(x)= & {} (1 + \alpha \xi + x)(x - \theta )\left( 1 - \frac{x}{\gamma }\right) , \end{aligned}$$

the above predator-prey system (8)–(9) can be expressed as

$$\begin{aligned} \dot{x}&= \left[ g(x) - y\right] f(x), \end{aligned}$$
(11)
$$\begin{aligned} \dot{y}&= \left[ \beta f(x)\left( 1 + \frac{\xi }{x}\right) - \delta \right] y. \end{aligned}$$
(12)

For the predator-prey system (11)–(12), the prey nullclines are: y-axis, the curve \(y = g(x)\) and the predator nullclines are: x-axis, the straight line \(x = \frac{\delta (1 + \alpha \xi ) - \beta \xi }{\beta - \delta }\), which is a part of \(\beta f(x)\left( 1 + \frac{\xi }{x}\right) - \delta = 0\) lying within the first quadrant. The prey nullcline \(y = g(x)\) intersects the x-axis at \((\theta , 0)\), \((\gamma , 0)\) and g(x) is positive in the interval \((\theta , \gamma )\). Also, the prey nullcline \(y = g(x)\) has a hump at \(x = x_H \in (\theta , \gamma )\) and it is increasing (ie, \(g^{\prime }(x) > 0\)) in \((\theta , x_H)\) and decreasing (ie, \(g^{\prime }(x) < 0\)) in \((x_H, \gamma )\) where

$$\begin{aligned} x_H= & {} \frac{1}{3}\left( (\gamma + \theta )--(1+\alpha \xi )\right) \\&+ \frac{1}{3}\left( \sqrt{[(\gamma + \theta )--(1+\alpha \xi )]^2 + 3[(1+\alpha \xi )(\gamma + \theta )-\gamma \theta ]}\right) . \end{aligned}$$

Thus, the prey nullcline \(y = g(x)\) is negative in \((0,\theta )\) and positive in \((\theta , \gamma )\) in case of strong Allee effect whereas it is positive in \((0,\gamma )\) in case of weak Allee effect. The predator nullcline \(x = \frac{\delta (1 + \alpha \xi ) - \beta \xi }{\beta - \delta }\) lies in the positive quadrant if \(\delta (1 + \alpha \xi ) - \beta \xi \) and \(\beta - \delta \) have the same sign.

3 Local stability and bifurcation

In this section we study the predator-prey system (11)–(12) to identify its equilibrium solutions, find their stability nature and provide local bifurcation analysis.

3.1 Existence of equilibria

First we find the number of equilibrium solutions admitted by the system (11)–(12) and investigate their local stability properties. We observe that the system (11)–(12) always admits the trivial equilibrium \(E_0 = (0, 0)\) and the axial equilibrium point \(E_{\gamma } = (\gamma , 0)\). It admits yet another axial equilibrium \(E_{\theta } = (\theta , 0)\) if the Allee effect in the prey population is strong. Existence of interior equilibrium \(E_* = (x_*, y_*)\), where

$$\begin{aligned} x_*= & {} \frac{\delta (1 + \alpha \xi ) - \beta \xi }{\beta - \delta },\nonumber \\ y_*= & {} g(x_*) = (1 + \alpha \xi + x_*)(x_* - \theta )\left( 1 - \frac{x_*}{\gamma }\right) , \end{aligned}$$
(13)

depends on the positivity of its components. Positivity of \(x_*\) is assured if one of the following conditions

$$\begin{aligned} \frac{\xi }{1+\alpha \xi }&< \frac{\delta }{\beta } < 1, \end{aligned}$$
(14)
$$\begin{aligned} \frac{\xi }{1+\alpha \xi }&> \frac{\delta }{\beta } > 1 \end{aligned}$$
(15)

is satisfied. Positivity of \(y_*\) is guaranteed if \(\theta < x_* < \gamma \) in case of strong Allee effect and \(0 < x_* < \gamma \) in case of weak Allee effect. Thus the considered system admits at least three and at most four equilibria incase of strong Allee effect and at least two and at most three equilibria in case of weak Allee effect.

In view of the above discussion we can conclude that, whenever the parameters satisfy either (14) or (15) and \(x_* < \gamma \), the system (11)–(12) admits a unique interior equilibrium if either

$$\begin{aligned} \theta (\beta - \delta ) < \delta (1+\alpha \xi ) - \beta \xi < \gamma (\beta - \delta ), \end{aligned}$$
(16)

or

$$\begin{aligned} 0 < \delta (1+\alpha \xi ) - \beta \xi < \gamma (\beta - \delta ), \end{aligned}$$
(17)

are satisfied in the cases of strong or weak Allee effect respectively.

Table 1 Table presenting the existence, eigenvalues of the Jacobian matrix (18) and stability nature at the concerned equilibrium point for the prey-predator system (11)–(12) with weak Allee effect (WAE) and strong Allee effect (SAE)
Full size table

3.2 Local stability analysis

Having identified the equilibrium solutions of the considered system (11)–(12), we evaluate the local stability nature of each of the equilibrium solutions. This enables us to understand the nonlinear dynamics associated with the system by investigating the involved local and global bifurcations. In order to study the bifurcations we concentrate on two parameters of the system given by \(\theta \), the strength of the Allee effect and \(\delta \), the death rate of the predator. Variations in the parameter \(\theta \) facilitates a comparative study in the system dynamics with respect to the strength of the Allee effect. Study associated with variations in the parameter \(\delta \) highlights dependance of the system dynamics on the death rate of the predators and also indicates the consequences of linear proportionate harvesting in predator population. Thus we derive the Jacobian matrix of the system (11)–(12) through the standard process of linearization and analyse its eigenvalues to identify the nature of the associated equilibrium point. The general form of the Jacobian matrix of the considered system (11)–(12) at any point (x, y) is given by

$$\begin{aligned}&J(x,y) \nonumber \\&\quad = \left( \begin{array}{cc} g^{\prime }(x)f(x) + [g(x)-y]f^{\prime }(x) &{} -f(x) \\ \beta y\left[ f^{\prime }(x)\left( 1+\frac{\xi }{x}\right) -\frac{f(x)}{x^2}\xi \right] &{} \quad \beta f(x)\left( 1+\frac{\xi }{x}\right) - \delta \\ \end{array} \right) \nonumber \\ \end{aligned}$$
(18)

where

$$\begin{aligned} f^{\prime }(x)= & {} \frac{1 + \alpha \xi }{(1 + \alpha \xi + x)^2},\\ g^{\prime }(x)= & {} -\frac{3}{\gamma }x^2 + 2\left[ \left( 1+\frac{\theta }{\gamma }\right) - \frac{1}{\gamma }(1+\alpha \xi )\right] x \\&+ \left[ (1+\alpha \xi )\left( 1+\frac{\theta }{\gamma }\right) - \theta \right] . \end{aligned}$$

Evaluating the Jacobian J(x, y) at each of the equilibrium solutions we come across four critical values for the parameter \(\delta \) given by \(\delta _{0} = \frac{\beta \xi }{1 + \alpha \xi }\), \(\delta _{T_{\theta }} = \frac{\beta (\theta + \xi )}{1 + \alpha \xi + \theta }\), \(\delta _{T_{\gamma }} = \frac{\beta (\gamma + \xi )}{1 + \alpha \xi + \gamma }\) and \(\delta _{H} = \frac{\beta (x_H + \xi )}{1 + \alpha \xi + x_H}\). Significance of these critical values is presented in Table 1.

From Table 1 we clearly understand that the system undergoes trans-critical bifurcation at the equilibria \(E_{\theta }\), \(E_{\gamma }\) (if admitted by the system) whenever the parameter \(\delta \) equals \(\delta _{T_{\theta }}\), \(\delta _{T_{\gamma }}\) respectively and Hopf-bifurcation occurs around \(E_*\) whenever \(\delta = \delta _{H}\), provided the parameters satisfy the condition (14). No Hopf bifurcation can occur at \(E_*\) if the parameters follow the condition (15), in which case \(E_*\) remains as a saddle point throughout. Below we give theoretical support for the occurrence of foresaid bifurcations.

3.3 Transcritical bifurcations

The expressions for the eigenvalues presented in Table 1 and the corresponding stability nature associated with the axial equilibria clearly indicate the occurrence of transcritical bifurcation at the equilibrium \(E_{\gamma }\) which is independent of sign of \(\theta \) and also at \(E_{\theta }\) for the case of strong Allee effect. An application of Sotomayer theorem (See “Appendix”) confirms the occurrence of transcritical bifurcation at \(E_{\gamma }\) and \(E_{\theta }\) when \(\delta = \delta _{T_{\gamma }}\) and \(\delta = \delta _{T_{\theta }}\) respectively.

Fig. 1
figure 1

Figure presenting the bifurcation diagram in the upper half plane of \((\theta , \delta )\) space for the case where parameters of the model (11)–(12) satisfy the condition (14). The curve in cyon represents (21). The transcritical bifurcation curve at \(E_{\theta }\) (22) is represented by the magenta curve. The green curve represents transcritical bifurcation curve at \(E_{\gamma }\) (23). The blue curve represents Hopf bifurcation curve (24) at \(E_*\). The curve presented in red colour represents occurrence of heteroclinic bifurcation between the two saddles \(E_{\theta }\) and \(E_{\gamma }\). (Color figure online)

Full size image

The interior equilibrium emerged or annihilated due to occurrence of two transcritical bifurcations at the axial equilibria \(E_{\theta }\) and \(E_{\gamma }\) respectively and the respective bifurcation conditions are given by (14) and (15). These two transcritical bifurcation curves are shown in the bifurcation diagrams presented in Figs. 1 and 4.

3.4 Hopf bifurcation

Let \(J_*\) represent the Jacobian matrix evaluated at \(E_*\). It can be easily verified that

$$\begin{aligned} Tr(J_*) = g^{\prime }(x_*)f(x_*) \end{aligned}$$
(19)

and

$$\begin{aligned} Det(J_*)= & {} \beta g(x_*)f(x_*)\left[ f^{\prime }(x_*)(1+\frac{\xi }{x_*})-\frac{f(x_*)}{x_*^2}\xi \right] \nonumber \\= & {} \frac{\beta g(x_*)f(x_*)}{(1 + \alpha \xi + x_*)^2}[1 + \alpha \xi - \xi ]. \end{aligned}$$
(20)

Clearly, Hopf bifurcation can not occur at \(E_*\) under the assumption (15), as it remains a saddle due to negativity of \(Det(J_*)\). Now, let us assume that the parameters satisfy the condition (14). Here we have \(Det(J_*)\) to be positive and \(Tr(J_*)|_{\delta = \delta _H} = f(x_H)g^{\prime }(x_H) = 0\) (since \(x_* = x_H\) whenever \(\delta = \delta _H\)). Differentiating trace of \(J_*\) with respect to the bifurcation parameter \(\delta \), we obtain

$$\begin{aligned} \frac{d}{d\delta }\left( Tr(J_*)\right)= & {} f^{\prime }(x_*)g^{\prime }(x_*)\frac{dx_*}{d\delta } + f(x_*)g^{\prime \prime }(x_*)\frac{dx_*}{d\delta }\\= & {} \left( f^{\prime }(x_*)g^{\prime }(x_*) + f(x_*)g^{\prime \prime }(x_*)\right) (1+\alpha \xi +x_*). \end{aligned}$$

Therefore

$$\begin{aligned}&\frac{d}{d\delta }\left( Tr(J_*)\right) |_{\delta = \delta _H} \\&\quad = \left( f^{\prime }(x_H)g^{\prime }(x_H) + f(x_H)g^{''}(x_H)\right) (1+\alpha \xi +x_H)\\&\quad = \frac{2x_H}{\gamma }\left( \sqrt{[(\gamma + \theta )-(1+\alpha \xi )]^2 + 3[(1+\alpha \xi )(\gamma + \theta )-\gamma \theta ]}\right) . \end{aligned}$$

Thus we have \(\frac{d}{d\delta }\left( Tr(J_*)\right) |_{\delta = \delta _H} > 0\). Hence the transversality condition for Hopf bifurcation is satisfied. The stability of the Hopf bifurcating limit cycle is determined by the first Lyapunov coefficient and we are not providing its expression here from the sake of brevity [46]. Numerically we have verified that the limit cycles are stable whenever they exist. Thus the considered system (11)–(12) undergoes a supercritical Hopf bifurcation at \(\delta = \delta _H\) whenever the parameters satisfy the condition (14) and this is independent of sign of \(\theta \). Here, we see that lowering \(\delta \) gives an increasing period of the limit cycle that arises through the Hopf-bifurcation. This Hopf-bifurcation curve \((\delta = \delta _H)\) is represented by the blue curve in Fig. 1. In case of strong Allee effect, the limit cycle, that appears through Hopf bifurcation, increase in size (as the predator isocline crosses the hump of the prey isocline and moves closer to y-axis) until it disappears through the heteroclinic bifurcation. At the heteroclinic bifurcation threshold we find the heteroclinic cycle joining the saddle points \(E_{\theta }\) and \(E_{\gamma }\). The curve on which the limit cycle disappears is represented by the red curve in Fig. 1.

4 Global dynamics

Information on the number of equilibrium solutions admitted by the system, the nature of the equilibria and the local bifurcations at equilibrium points identify the following four significant relations between the system parameters using which the global dynamics of the system can be determined.

$$\begin{aligned} h_1(\theta , \delta )\equiv & {} \beta \xi - \delta (1+\alpha \xi ) = 0, \end{aligned}$$
(21)
$$\begin{aligned} h_2(\theta , \delta )\equiv & {} \beta (\theta + \xi ) - \delta (1+\alpha \xi + \theta ) = 0, \end{aligned}$$
(22)
$$\begin{aligned} h_3(\theta , \delta )\equiv & {} \beta (\gamma + \xi ) - \delta (1+\alpha \xi + \gamma ) = 0 \end{aligned}$$
(23)

and

$$\begin{aligned} h_4(\theta , \delta ) \equiv g^{\prime }(x_*) = g^{\prime }\Big (\frac{\delta (1 + \alpha \xi ) - \beta \xi }{\beta - \delta }\Big ) = 0. \end{aligned}$$
(24)

Observe that the above curves represent occurrence of transcritical bifurcation at \(E_0\), \(E_{\theta }\), \(E_{\gamma }\) and Hopf-bifurcation at \(E_{*}\) respectively. Since we assumed \(\theta \) and \(\delta \) to be bifurcation parameters, we plot the above curves in the upper half plane of \((\theta , \delta )\) space for fixed parameter values of \(\gamma \), \(\alpha \), \(\xi \) and \(\beta \). Here we obtain two distinct bifurcation diagrams (Figs. 1, 4) corresponding to the parametric restrictions satisfying the conditions (14) and (15) respectively. Apart from the curves (21)–(24), Fig. 1 presents yet another curve (shown in red colour) that signifies the occurrence of saddle-saddle connection in the phase space. Thus, there are five curves in Fig. 1 and three in Fig. 4. In both these figures, the first and second quadrants respectively represent the system bifurcations associated with strong and weak Allee effects respectively.

Note that the curves presented in each of the Figs. 1 and 4 divide the two quadrants into finite number of subregions. These subregions characterise nature of the equilibria admitted by the system. Observe that there are ten such regions (six for strong Allee effect and four for weak Allee effect) in Fig. 1 and seven such regions (four for strong Allee effect and three for weak Allee effect) in Fig. 4. Below we shall discuss the global dynamics of the system as the bifurcation parameters move through each of these regions.

Fig. 2
figure 2

Phase portraits of the considered system (11)–(12) satisfying (14) with strong Allee effect for different values of \(\delta \). \(\delta = 0.25\) (Region \(R_1\)), \(\delta = 0.55\) (Region \(R_2\)), \(\delta = 0.62\) (Region \(R_3\)), \(\delta = 0.6746878728994055\) (Heteroclinic), \(\delta = 0.678\) (Region \(R_4\)), \(\delta = 0.74\) (Region \(R_5\)), \(\delta = 0. 81\) (Region \(R_6\)). Here \(\theta = 0.2\), \(\gamma = 1.2\), \(\alpha = 4.3\), \(\xi = 1.4\) and \(\beta = 2.5\) are fixed parameters

Full size image
Fig. 3
figure 3

Phase portraits of the considered system (11)–(12) with weak Allee effect for different values of \(\delta = 0.81\) (Region \(R_7\)), \(\delta = 0.725\) (Region \(R_8\)), \(\delta = 0.66\) (Region \(R_9\)), \(\delta = 0.4\) (Region \(R_{10}\)). Here \(\theta = -0.75\), \(\gamma = 1.2\), \(\alpha = 4.3\), \(\xi = 1.4\) and \(\beta = 2.5\) are fixed parameters

Full size image
Fig. 4
figure 4

Figure presenting the bifurcation diagram in the upper half plane of \((\theta , \delta )\) space for the case where parameters of the model (11)–(12) satisfying (15). The curve in cyon represents (21). The transcritical bifurcation curve at \(E_{\theta }\) (22) is represented by the magenta curve. The green curve represents transcritical bifurcation curve at \(E_{\gamma }\) (23). (Color figure online)

Full size image

Let us consider the bifurcation diagram presented in Fig. 1. In the case of strong Allee effect, we start with a small value of \(\delta \) lying in the region \(R_1\). In this region we have \(h_1(\theta , \delta ) < 0\), \(h_2(\theta , \delta ) > 0\) and \(h_3(\theta , \delta ) > 0\) meaning that \(E_0\) is saddle, \(E_{\theta }\) is unstable node and \(E_{\gamma }\) is saddle point respectively and hence all the solutions starting in this region exhibit blow up in the predator component (see phase portraits for Region \(R_1\) in Fig. 2). Essentially the region \(R_1\) represents the situation where the predator nullcline lies in the second quadrant of the phase space. As we move from \(R_1\) to \(R_2\) we have \(h_1(\theta , \delta ) > 0\), \(h_2(\theta , \delta ) > 0\) and \(h_3(\theta , \delta ) > 0\) indicating that \(E_0\) is stable, \(E_{\theta }\) is unstable node and \(E_{\gamma }\) is saddle point respectively. Thus the predator and prey population always tend to extinction. Therefore \(E_0\) is globally asymptotically stable (see phase portraits for Region \(R_2\) in Fig. 2).

As we enter \(R_3\) from \(R_2\) we have emergence of the interior equilibrium point \(E_*\). In this region we have \(h_2(\theta , \delta ) < 0\) and \(h_4(\theta , \delta ) > 0\). Hence we have \(E_{\theta }\) to be saddle point and the interior equilibrium point \(E_*\) to be unstable node. The transcritical bifurcation taking place due to exchange of stability between \(E_{\theta }\) and \(E_*\) along the curve (23) that separates the region \(R_2\) and \(R_3\). The nature of the equilibrium points \(E_0\) and \(E_{\gamma }\) remains unchanged. \(E_0\) being the only asymptotically stable equilibrium, all the non equilibrium solutions initiating in the positive quadrant of the phase space approach \(E_0\) asymptotically leading to eventual extinction of both predator and prey population (see phase portraits for Region \(R_3\) in Fig. 2).

As we enter the region \(R_4\) from \(R_3\) the interior equilibrium point \(E_*\) turns into unstable focus from unstable node and the stability nature of all other equilibria remains unchanged. The heteroclinic orbit (green colour path), joining the two saddle points \(E_{\theta }\) and \(E_{\gamma }\), that occurs along the boundary curve between \(R_4\) and \(R_3\) is presented in the bifurcation diagram (see Heteroclinic, Fig. 2). Since this saddle-saddle connection is broken in \(R_4\), the stable manifold of \(E_{\theta }\) divides the phase plane into two invariant sub regions such that one of them contains \(E_0\) and the other contains \(E_*\) and \(E_{\gamma }\). All the paths with their initial state present in the region containing \(E_0\) will eventually reach \(E_0\) and those initiating in the region containing \(E_*\) and \(E_{\gamma }\) approach a stable limit cycle surrounding the unstable focus \(E_*\) (see phase portraits for Region \(R_4\) in Fig. 2). Therefore co-existence or extinction of population depends on the initial state of the system. As we move from \(R_4\) to \(R_5\) we have \(h_4(\theta , \delta ) < 0\) and hence the interior equilibrium \(E_*\) becomes asymptotically stable while the stability nature of the axial equilibria \(E_0\), \(E_{\theta }\) and \(E_{\gamma }\) remain unchanged. Here the separatrix (stable manifold of \(E_{\theta }\)) marks the boundary between the basins of attraction of the two asymptotically stable equilibria \(E_0\) and \(E_*\). Depending on the initial state in the basins of attraction, the trajectories of the system either tend to \(E_0\) or to \(E_*\). So, either the predator and prey go to extinction or there is stable coexistence between the prey and the predator (see phase portraits for Region \(R_5\) in Fig. 2). Thus as we move from \(R_5\) to \(R_4\), the limit cycles that appear due to hopf bifurcation occurring along the curve \(h_4(\theta , \delta ) = 0\) increase in size until they disappear when the heteroclinic curve joining the points \(E_{\theta }\) and \(E_{\gamma }\) is broken, which occurs on the boundary between \(R_4\) and \(R_3\).

As we enter the region \(R_6\) from \(R_5\) we have occurrence of transcritical bifurcation at \(E_{\gamma }\) leading to exchange of stability between \(E_*\) and \(E_{\gamma }\) and disappearance of \(E_*\). The system admits two asymptotically stable equilibria \(E_0\) and \(E_{\gamma }\) whose basins of attraction are separated by the stable manifold of the saddle point \(E_{\theta }\). Hence, the eventual state of the system (\(E_0\) or \(E_{\gamma }\)) depends on the initial state. Irrespective of the initial state, the predators get extinct where as the prey population either survives and reaches its carrying capacity or gets extinct (see phase portraits for Region \(R_6\) in Fig. 2).

So far we have discussed the global dynamics of the considered system (11)–(12) under the influence of strong Allee effect with the assumption that the parameters satisfy the condition (14). Now let us look at the bifurcation diagram for the case of weak Allee effect which is represented by the second quadrant in the Fig. 1. In region \(R_7\), we have that \(h_1(\theta , \delta ) < 0\) and \(h_3(\theta , \delta ) < 0\) meaning that \(E_0\) is saddle and \(E_{\gamma }\) is asymptotically stable (see phase portraits for Region \(R_7\) in Fig. 3). Hence all the solutions initiating in the positive quadrant eventually approach \(E_{\gamma }\) making it globally asymptotically stable. As we enter \(R_8\) from \(R_7\) through the curve \(h_3(\theta , \delta ) = 0\), transcritical bifurcation takes place at \(E_{\gamma }\) leading to exchange of stability between \(E_{\gamma }\) and the newly emerged interior equilibrium \(E^*\). Since \(h_3(\theta , \delta ) > 0\) and \(h_4(\theta , \delta ) < 0\) we have \(E_{\gamma }\) to be saddle point and the interior equilibrium point \(E_*\) to be asymptotically stable node (see phase portrait for Region \(R_8\) in Fig. 3). Thus, in the region \(R_8\), \(E_*\) is globally asymptotically stable. As we move from \(R_8\) to \(R_9\) we have \(h_4(\theta , \delta ) > 0\) and hence the interior equilibrium point \(E_*\) is unstable focus surrounded by a limit cycle while the stability of other equilibrium points remains unchanged. Therefore both the predators and prey asymptotically approach the stable limit cycle surrounding the unstable focus (see phase portrait for Region \(R_9\) in Fig. 3). This is due to the occurrence of Hopf-bifurcation along the curve \(h_4(\theta , \delta ) = 0\). Here the limit cycle surrounding \(E_*\) is the global attractor. Moving from \(R_9\) to \(R_{10}\) we have \(h_1(\theta , \delta ) > 0\) and hence \(E_0\) becomes unstable and the interior equilibrium disappears. The saddle nature of the axial equilibrium \(E_{\gamma }\) remains unchanged. Therefore all solutions starting in the first quadrant blow up in the \(y-\)direction (see phase portraits for Region \(R_{10}\) in Fig. 3).

So far, we described the bifurcation diagram (Fig. 1) in greater detail by using phase portraits that give better understanding of the dynamical behavior of the considered model (11)–(12). The phase portraits corresponding to the different regions of the bifurcation diagram (Fig. 1) for strong Allee effect \((R_1 - R_6)\) and weak Allee effect \((R_7 - R_{10})\) are presented in Figs. 2,  3 respectively. The values of the parameters are mentioned in the captions of the respective figures. In these phase portraits, the saddle, stable, unstable equilibrium points are represented by circles, circles filled with blue color and circles filled with red color respectively. Stable and unstable manifolds are represented by red curves. Limit cycles are represented in green and paths are represented in blue.

Now Fig. 5 presents another bifurcation diagram (in \((x, \delta )\) plane) that depicts the stability nature of the equilibria of the considered model (11)–(12) with strong Allee effect under the parametric condition (14). The three blue color dotted vertical lines \(\delta = \delta _{T_{\theta }}\), \(\delta = \delta _H\) and \(\delta = \delta _{T_{\gamma }}\) represent transcritical bifurcation at \(E_{\theta }\), Hopf-bifurcation at \((x_H, g(x_H))\) and transcritical bifurcation at \(E_{\gamma }\) respectively. The heteroclinic orbit, connecting the two saddle points \(E_{\theta }\) and \(E_{\gamma }\) comes into existence at \(\delta = \delta _{Het}\). The axial equilibrium points always exist and \(E_0\) is asymptotically stable represented by solid magenta line, \(E_{\theta }\) is unstable (saddle) if \(\delta < \delta _{T_{\theta }}\) (\(\delta > \delta _{T_{\theta }}\)) and \(E_{\gamma }\) is saddle (stable) if \(\delta < \delta _{T_{\gamma }}\) (\(\delta > \delta _{T_{\gamma }}\)). The interior equilibrium point exists for \(\delta \in (\delta _{T_{\theta }}, \delta _{T_{\gamma }})\) but it is unstable node for \(\delta \in (\delta _{T_{\theta }}, \delta _{Het})\), unstable focus for \(\delta \in (\delta _{Het}, \delta _H)\) and locally asymptotically stable for \(\delta \in (\delta _H, \delta _{T_{\gamma }})\). Red color dotted line represents the x component of the unstable interior equilibrium point. The x component of the stable interior equilibrium point along with the minimum and maximum of the stable limit cycle surrounding an unstable focus is presented (red in color). One can easily observe the existence of bistability in the system for \(\delta > \delta _{Het}\) for the considered model (11)–(12) with strong Allee effect. Similarly, Fig. (6) presents the bifurcation diagram for the system (11)–(12) with weak Allee effect when the involved parameters satisfy the condition (14).

Fig. 5
figure 5

Figure presenting the bifurcation diagram for stability of equilibrium point of the considered model with strong Allee effect under the parametric conditions (14). \(E_0\) is stable and \(E_\theta \) is unstable for \(\delta \in [0.52,0.82]\). \(E_\gamma \) is unstable for \(\delta \in [0.52,\,\delta _{T_\gamma })\) and stable for \(\delta \in (\delta _{T_\gamma },\,0.82]\), \(E_\gamma \) and \(E_*\) coincides at the transcritical bifurcation threshold \(\delta _{T_\gamma }\). Interior equilibrium point \(E_*\) is feasible for \(\delta _{T_\theta } < \delta < \delta _{T_\gamma }\) and is stable for \(\delta _H < \delta < \delta _{T_\gamma }\). \(E_*\) loses stability through super-critical Hopf-bifurcation at \(\delta _H\) and stable limit cycle exists for \(\delta _{{\text {Het}}} < \delta < \delta _H\). The stable limit cycle disappear through a heteroclinic bifurcation and heteroclinic loop is formed by joining two axial equilibrium points \(E_\theta \) and \(E_\gamma \)

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Fig. 6
figure 6

Figure presenting the bifurcation diagram for stability of equilibrium point of the considered model with weak Allee effect where the parameters of the system satisfying (14). \(E_0\) is stable for \(\delta \in [0.48,0.88]\) and \(E_\gamma \) is stable for \(\delta \in (\delta _{T_\gamma },\,0.88]\). \(E_\gamma \) and \(E_*\) coincide with each other at the transcritical bifurcation threshold \(\delta _{T_\gamma }\). Interior equilibrium point \(E_*\) is feasible for \(\delta _{T_\theta }\,<\,\delta \,<\, \delta _0\) and is stable for \(\delta _H\,<\,\delta \,<\,\delta _{T_\gamma }\). \(E_*\) loses stability through super-critical Hopf-bifurcation at \(\delta _H\) and stable limit cycle exists for \(\delta _0\,<\,\delta \,<\,\delta _H\)

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Fig. 7
figure 7

Phase portraits of the considered system (11)–(12) with strong Allee effect for different values of \(\delta = 1.80\) (Region \(S_1\)), \(\delta = 2.2\) (Region \(S_2\)), \(\delta = 2.52\) (Region \(S_3\)), \(\delta = 2.824\) (Region \(S_4\)). Here \(\theta = 0.2\), \(\gamma = 1.2\), \(\alpha = 0.01\), \(\xi = 2.85\) and \(\beta = 1\) are fixed parameters

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Fig. 8
figure 8

Phase portraits of the considered system (11)–(12) with weak Allee effect for different values of \(\delta = 2.82\) (Region \(S_5\)), \(\delta = 2.52\) (Region \(S_6\)), \(\delta = 1.80\) (Region \(S_7\)). Here \(\theta = -0.75\), \(\gamma = 1.2\), \(\alpha = 0.01\), \(\xi = 2.85\) and \(\beta = 1\) are fixed parameters

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Now, let us look at the other case where parameters of the system follow the condition (15). Bifurcation diagram corresponding to this case is represented in Fig. 4 which presents division of parameter space into seven regions. The phase portraits of the system (11)–(12) corresponding to the subregions depicted in Fig. 4 are presented in the Figs. 7 and  8. The description pertaining to the bifurcation diagram and the corresponding phase portraits can be presented in a similar as in the previous case.

5 Discussion

Growing importance for eco-friendly practices for managing and controlling ecosystems has driven experimental scientists to perform various experiments to identify appropriate methods for biological control. One of the established methods happens to be provision of additional food to predators. This method is expected to enhance the survival of the predators in the environment. Further, it is anticipated that the increased growth in predators would control and limit the resident prey (pest). On the contrary, it is observed that provision of additional food to predators did not always increase the population density of the predators. There were occasions where this provision lead to decrease in predators and increase in the pest density. This contra intuitive observation has motivated the theoretical scientists to investigate into the influence of additional food on the dynamics of interacting species such as predator-prey systems in order to offer explanation for the observations made by experimental scientists. Recent studies [42–45] not only highlight the necessity for undertaking such studies but also provide state of art in this area.

In this article we study the dynamics of a predator-prey system when the predator is provided with additional food and the prey growth is influenced by Allee effect. The considered model takes both strong and weak Allee effects into consideration. Analysis of the system highlights the bifurcations associated with the dynamics. It is observed that inclusion of additional food into the system enriches the system dynamics. This analysis also offers strategies to manipulate the additional food provision for accomplishing the desired results in the system such as elimination of prey from the system (an application to pest management) or distraction of predators from prey (an application to biological conservation) there by enhance chances of survival for prey.

While provision of additional food to predators does not eliminate the characteristic features of the existing Allee effect in the system, it does affect the level of coexistence of the species considerably. The existence of interior equilibrium and variation in its stability behaviour much depends on the measure of a specific combination of quality and quantity of the additional food relative to unity as well as the ratio between the maximum growth rate and death rate of the predators. If the maximum growth rate of the predators is higher than their natural death rate then, with the quality and quantity of the additional food satisfying certain relations, it is possible to bring in coexistence of the species (if it is not originally present), alter the stability nature of the interior equilibrium and even eliminate the coexistence by changing the quality and quantity of additional food provided to predators.

On the other hand, if the maximum growth rate of the predators is lower than their natural death rate then although an interior equilibrium can be brought in (only in the case of strong Allee effect) through provision of additional food to predators, for all practical purposes co-existence of the species can not be achieved due to saddle nature of the interior equilibrium which can not be altered with variations in quality and quantity of additional food. As a result the system experiences either unlimited growth in the predator combined with prey extinction or asymptotic approach of the prey population to its carrying capacity along with extinction of predator population. Various regions presented in the bifurcation diagrams and the associated phase portraits can be used as the guiding principles for achieving the needed biological control results.