Transformation | |||
Non-dimensional system | where |
We propose and analyze the effects of a generalist predator-driven fear effect on a prey population by considering a modified Leslie-Gower predator-prey model. We assume that the prey population suffers from reduced fecundity due to the fear of predators. We investigate the predator-prey dynamics by incorporating linear, Holling type Ⅱ and Holling type Ⅲ foraging strategies of the generalist predator. As a control strategy, we have considered density-dependent harvesting of the organisms in the system. We show that the systems with linear and Holling type Ⅲ foraging exhibit transcritical bifurcation, whereas the system with Holling type Ⅱ foraging has a much more complex dynamics with transcritical, saddle-node, and Hopf bifurcations. It is observed that the prey population in the system with Holling type Ⅲ foraging of the predator gets severely affected by the predation-driven fear effect in comparison with the same with linear and Holling type Ⅱ foraging rates of the predator. Our model simulation results show that an increase in the harvesting rate of the predator is a viable strategy in recovering the prey population.
Citation: |
Figure 1. Mutual position of prey-nullclines (red) and predator-nullclines (blue) of the system (3) due to the changes in $ h_1 $ and $ h_2 $, other parameters are taken from Table 2. The system is LAS at $ (a) $ $ E^* $ ($ E_0 $, $ E_1 $ and $ E_2 $ are unstable), $ (b) $ $ E_2 $ ($ E_0 $ and $ E_1 $ are unstable; $ E^* $ does not exist), $ (c) $ $ E_1 $ ($ E_0 $ is unstable; $ E^* $ and $ E_2 $ do not exist) and $ (d) $ $ E_0 $ ($ E^* $, $ E_1 $ and $ E_2 $ do not exist)
Figure 3. Two-parameter bifurcation plots with the bifurcation parameters $ (a) $ $ h_1 $ and $ h_2 $, $ (b) $ $ h_1 $ and $ \beta $, $ (c) $ $ h_2 $ and $ \beta $, $ (d) $ $ \beta $ and $ \eta_1 $, $ (e) $ $ h_1 $ and $ \eta_1 $, $ (f) $ $ h_2 $ and $ \eta_1 $. All other parameters are taken from Table 2. The coloured bars represent the prey population density
Figure 4. Mutual position of prey-nullclines (red) and predator-nullclines (blue) of the system (4) due to the changes in $ h_1 $ and $ h_2 $, all other parameters are taken from Table 2. $ (a) $ The system is LAS at the unique interior equilibrium $ E^*_1 $ ($ E_0 $, $ E_1 $ and $ E_2 $ are unstable). $ (b) $ The system has bistability at $ E_2 $ and $ E^*_1 $ ($ E_0 $, $ E_1 $ and $ E^*_2 $ are unstable). The system is LAS at $ (c) $ $ E_2 $ ($ E_0 $ and $ E_1 $ are unstable; $ E^*_i $ does not exist), $ (d) $ $ E_2 $ ($ E_0 $ is unstable; $ E^*_i $ and $ E_1 $ do not exist), $ (e) $ $ E_1 $ ($ E_0 $ is unstable; $ E^*_i $ and $ E_2 $ do not exist) and $ (f) $ $ E_0 $ ($ E^*_i $, $ E_1 $ and $ E_2 $ do not exist) $ (i = 1,2) $
Figure 5. One-parameter bifurcation plots of the system (4) due to the changes in $ h_1 $ $ (a) $ for $ h_2<r $ and $ \eta_1 = 0.05 $, where a transcritical bifurcation occurs at $ h_1^{**} = 0.6585 $; $ (b) $ for $ h_2<r $ and $ \eta_1 = 0.125 $, a transcritical and a saddle-node bifurcation occur at $ h_1^{**} = 0.1475 $ and $ h_{1cr}^- = 0.3685 $ respectively. One-parameter bifurcation plots of the system (4) due to the changes in $ h_2 $ $ (c) $ for $ h_1<h_1^* $ and $ \eta_1 = 0.05 $, where a transcritical bifurcation occurs at $ h_2^{**} = 0.3495 $; $ (d) $ for $ h_1<h_1^* $ and $ \eta_1 = 0.125 $, a transcritical and a saddle-node bifurcation occur at $ h_2^{**} = 0.74 $ and $ h_{2sn} = 0.64 $ respectively
Figure 7. Two-parameter bifurcation plots with the bifurcation parameters $ (a) $ $ h_1 $ and $ \beta $, $ (b) $ $ h_2 $ and $ \beta $, $ (c) $ $ h_1 $, and $ \eta_1 $, $ (d) $ $ h_2 $ and $ \eta_1 $; other parameter values are taken from Table 2
Figure 8. $ (a) $ For $ \alpha = 0.4993 $, the phase space shows the existence of stable (in green) and unstable (in red) manifolds of the system (4). The unstable limit cycle around $ E^*_1 $ is represented in blue. The curves representing the changes in $ (b) $ $ {\rm{Tr}}(J^*_1) $, $ {\rm{Det}}(J^*_1) $ and $ (c) $ $ \frac{d}{d\alpha}{\rm{Tr}}(J^*_1) $ due to the changes in $ \alpha $ verify the occurrence of a Hopf bifurcation of the system (4) at $ \alpha = 0.4993 $
Figure 9. Mutual position of prey-nullclines (red) and predator-nullclines (blue) of the system (5) due to the changes in $ h_1 $ and $ h_2 $, other parameters are taken from Table $ 1 $. The system is LAS at $ (a) $ $ E_* $ ($ E_0 $, $ E_1 $ and $ E_2 $ are unstable), $ (b) $ $ E_2 $ ($ E_0 $ and $ E_1 $ are unstable; $ E_* $ does not exist), $ (c) $ $ E_1 $ ($ E_0 $ is unstable; $ E_* $ and $ E_2 $ do not exist) and $ (d) $ $ E_0 $ ($ E_* $, $ E_1 $ and $ E_2 $ do not exist)
Figure 10. One-parameter bifurcation plots of the system (5) due to the changes in $ (a) $ $ h_1 $ where $ h_2<r $, $ (b) $ $ h_2 $ where $ h_1<h_1^{\#} $. $ (c) $ A two-parameter bifurcation plot with $ h_1 $ and $ h_2 $ as bifurcation parameters, where $ h_1 = 1 $, $ h_1 = h_1^{\#} $ and $ h_2 = r $ are transcritical bifurcation curves. All other parameters are taken from Table 2
Figure 11. Two-parameter bifurcation plots of the system (5) with the bifurcation parameters $ (a) $ $ h_1 $ and $ \beta $, $ (b) $ $ h_2 $ and $ \beta $, $ (c) $ $ h_1 $ and $ \eta_1 $, $ (d) $ $ h_2 $ and $ \eta_1 $, where $ h_1 = h_1^{\#} $, $ h_2 = h_2^{\#} $, and $ h_2 = r $ are transcritical bifurcation curves
Figure 12. Local sensitivity of the prey response for $ (a) $ linear, $ (b) $ Holling type Ⅱ, and $ (c) $ Holling type Ⅲ foraging of the predator. For comparison, model simulations before parameter manipulations, are shown in black line. The prey density from simulations where particular parameter values were increased by $ 10\% $ are shown in red lines, as are the prey density from simulations where particular parameter values were decreased by $ 10\% $ in blue lines
Figure 15. Local sensitivity of the predator response for $ (a) $ linear, $ (b) $ Holling type Ⅱ, and $ (c) $ Holling type Ⅲ foraging of the predator. For comparison, model simulations before parameter manipulations, are shown in black line. The predator density from simulations where particular parameter values were increased by $ 10\% $ are shown in red lines, as are the predator density from simulations where particular parameter values were decreased by $ 10\% $ in blue lines
Table 1. Non-dimensionalized system
Transformation | |||
Non-dimensional system | where |
Table 2. Tables of parameter values
(a) | ||
Original parameters | ||
Parameter | Description | Value |
$R_1$ | Intrinsic growth rate of prey | 0.03 |
$R_2$ | Intrinsic growth rate of predator | 0.03 |
$B$ | The level of fear | 4 |
$A_1$ | Consumption rate of predator | 0.5 |
$A_2$ | Intraspecific competition of predator | 0.5 |
$K$ | Carrying capacity of prey | 2 |
$\eta$ | Alternative prey density | 0.25 |
$B_1$ | Half saturation coefficient | 0.1 |
$H_1$ | Harvesting rate of prey | 0.01 |
$H_2$ | Harvesting rate of predator | 0.02 |
(b) | ||
Non-dimensional parameters | ||
Parameter | Value | |
$r$ | 1 | |
$\alpha$ | 0.03 | |
$\beta$ | 0.48 | |
$\eta_1$ | 0.125 | |
$b$ | 0.025 | |
$h_1$ | 0.333 | |
$h_2$ | 0.667 |
Table 3. Existence and local stability of equilibria of system (3)
Equilibria | Sufficient condition for existence | Local asymptotic stability |
Always | ||
Table 4. Existence and local stability of equilibria of system (4)
Equilibria | Sufficient condition for existence | Local asymptotic stability |
Always | ||
Table 5. Existence and local stability of equilibria of system (5)
Equilibria | Sufficient condition for existence | Local asymptotic stability |
Always | ||
Table 6. Comparison of the critical threshold values for transcritical bifurcation (TB) and saddle-node bifurcation (SNB) of the three systems
Bifurcation parameter | Linear | Holling type Ⅱ | Holling type Ⅲ | |||
Threshold | Bifurcation | Threshold | Bifurcation | Threshold | Bifurcation | |
$h_1$ ($h_2 < r$) |
$h_1^*=0.8971$ | TB | $h_1^{**}=0.1475$ $h_{1sn}^-=0.3685$ |
TB SNB |
$h_1^{\#}=0.79$ | TB |
$h_1$ ($h_2>r$) |
$h_1^*=1$ | TB | $h_1^{**}=1$ | TB | $h_1^{\#}=1$ | TB |
$h_2$ ($h_1 < 1$) |
$h_2^*=1$ | TB | $h_2^{**}=0.74$ $h_{2sn}=0.64$ |
TB SNB |
$h_2^{\#}=1$ | TB |
$\beta$ ($h_1 < 1$ & $h_2 < r$) |
$\beta^*=16.8$ | TB | $\beta^{**}=16$ $\beta_{sn}=18.51$ |
TB SNB |
$\beta^{\#}=1.5$ | TB |
$\eta_1$ ($h_1 < 1$ & $h_2 < r$) |
$\eta_1^*=0.825$ | TB | $\eta_1^{**}=0.2$ $\eta_{1sn}=0.441$ |
TB SNB |
$\eta_1^{\#}=0.3721$ | TB |
Table 7.
Bifurcation parameters with different foraging types and corresponding basins of attraction at
Parameters | Largest basin of recovery | Smallest basin of recovery |
Linear | Holling type Ⅱ | |
Linear | Holling type Ⅲ | |
Holling type Ⅲ | Holling type Ⅱ | |
Linear | Holling type Ⅱ | |
Linear | Holling type Ⅱ |
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Mutual position of prey-nullclines (red) and predator-nullclines (blue) of the system (3) due to the changes in
One-parameter bifurcation plots of the system
Two-parameter bifurcation plots with the bifurcation parameters
Mutual position of prey-nullclines (red) and predator-nullclines (blue) of the system (4) due to the changes in
One-parameter bifurcation plots of the system (4) due to the changes in
Two-parameter bifurcation plots with
Two-parameter bifurcation plots with the bifurcation parameters
Mutual position of prey-nullclines (red) and predator-nullclines (blue) of the system (5) due to the changes in
One-parameter bifurcation plots of the system (5) due to the changes in
Two-parameter bifurcation plots of the system (5) with the bifurcation parameters
Local sensitivity of the prey response for
One-parameter bifurcation plots due to the changes in
One-parameter bifurcation plots due to the changes in
Local sensitivity of the predator response for