It is confirmed that the work contained in this thesis entitled "Dynamics and control of a predator-prey system with the supply of additional food for predators" was done by me, under the supervision of Dr. This thesis examines the efficient and important role of supplementary food in a predator-prey system.
Introduction
Taking this into account, we present a modified version of the classic ratio-dependent predator-prey model, by including the provision of additional food to the predator population. A predator-prey system with logistic growth but non-linear diffusion for the predator population is studied in [34].
Thesis Outline
Model with Additional Food
We now present a modified ratio-dependent model that incorporates the provision of additional food to the predators. Therefore, the total number of prey per predator per time unit in the presence of additional food supply for predators.
Non-Dimensionalized Form of the Model
- Ratio-Dependent Model
- Modified Ratio-Dependent Model
- Existence of Equilibrium Points
- Boundedness of the Solution
- Local Stability Analysis
- Global Stability Analysis
In this chapter, the dynamics of the modified ratio-dependent model are analyzed in terms of local and global stability analysis. The conditions on the model parameters, required for the existence of the three equilibrium points, are given in Table 3.1.
Modified Ratio-Dependent Model
- Existence of Equilibrium Points
- Boundedness of the Solution
- Local Stability Analysis
- Case 1
- Case 2
- Case 3
- Limit Cycle
- Global Stability Analysis
The location of the existence/non-existence of the equilibria in (α, ξ)-parameter space is given in Figure 3.1 (for the existential state of the internal equilibria see Appendix A). The axial equilibrium B(1,0) is globally stable, if any of the following results are true. a) b≥1 and the axial equilibrium point B(1,0) is locally asymptotically stable.
Observations, Ecological Interpretations and Justifications of the Results
Some Other Observations
It is observed that when the quality of supplementary food is high (lower value of α), the C balance is stable and ensures the survival of the predator population despite the absence of prey. Decreasing quality of supplementary food now causes prey and predator populations to converge towards carrying capacity or extinction. Keeping the value of α close to 1−1c. b), an increase in the value of ξ results in an increase in the predator population, while the prey population is small (relative to the population at α value away from 1−1c.
In the figure, (a) →(b)→ (c) or (d) → (e)→ (f), indicates that the stability of equilibrium C is moving towards the stability of equilibrium B as we increase the value of α for fixξ. a)→(d) or (b)→(e), shows that the size of predators is increasing as we increase the value of ξ for fixed α.
Discussion and Conclusion
On the other hand, the analysis of the modified ratio-dependent model explicitly shows the existence of a globally stable equilibrium point (0,(b−α)ξ) and here, too, predator growth is proportional to the amount of additional food, which addresses this issue more realistically. This model offers us a way to achieve greater control over the dynamics of the system by appropriately manipulating the quantity and quality of additional food supplied to the predator in the system. Parameters D1 and D2 are the diffusivity of the prey and predator density, respectively.
While α is representative of the quality (nutritional value) of the additional food compared to that of prey, ξ symbolizes the amount of additional food provided to the predators.
The Model without Diffusion
The no-flow boundary condition is used to facilitate the study of the self-organizing patterns and this condition places restriction on any inflow/outflow into/out of the domain [50]. We now non-dimensionalize the above system using the transformations, x = N/K, y = P/Ke1h1 and t=rT, to obtain,. Here are the parameters that appear in the original relationship-dependent model, namely c,mand bare ecological in nature.
However, the parameters α and ξ appear due to additional food supply and can be seen as parameters for biological control.
Turing Instability
Now the eigenvalue σjk, which determines the time growth, is given by the roots of the characteristic polynomial. So, for Turing instability to occur, the condition det(Ji−λjD) ≤ 0 must be satisfied for somej≥1. Thus, a necessary condition for inhomogeneous spatial patterns is that predators disperse faster than prey.
In other words, the prey and the predator play the role of activator and inhibitor, respectively.
Bifurcations and Turing Space
As for the Turing bifurcation curve T b , in the region above T b and bounded by T cb1 and T cb2 in Figure 4.1, (x2, y2) is Turing stable. Similarly, the changes in the Turing stability of (x2, y2) over T b are identical to the previous case. The Turing space is the region bounded by the bifurcation curves T b and Hopf, as well as the saddle-knot curve.
In this case, as seen in Figure 4.3, the Turing space is the region bounded by the saddle T band curves.
Turing Patterns
The simulation was run over a large number of time steps until a steady-state Turing pattern was reached. We see that a steady state occurs at time t = 1000, where there are regions of high (red) and low density (blue). In this case, however, the simulations had to be run for a long time before they reached a steady state.
The emergence of patterns is similar to that of the preceding case, except that the steady state Turing pattern is reached much later.
Conclusion
In this chapter we present two models that include prey harvesting in the classic and the modified ratio-dependent predator-prey model. We investigate the consequences of providing additional food (as part of the total harvesting effort) to predators in prey harvesting. To begin with, we consider a general class of predator-prey systems, which will include all three predator-prey systems that will be discussed in this chapter, namely the ratio-dependent system with prey harvesting and the modified ( by way of provision) of additional food to the predators) ratio-dependent system with prey harvesting.
We will show that the solution for this class of predator-prey system (as given below) is bounded in the first quadrant of R2.
Stability of the Classical Ratio-Dependent Model with Prey Harvesting
Existence of Equilibrium Points
By means of the analysis of the vector field it can be shown that the solutions of the general class of the system (5.1.1) are non-negative. The equilibrium (0,0) represents the extinction of both species and (1,0) represents the extinction of the predator population with the prey population at its carrying capacity. It can be easily observed from the internal equilibria of the system (2.1.2) and (5.2.1), that harvesting the prey results in a decrease in the equilibrium level of both the prey and the predator, due to the increase in the mortality rate hunt. population and thus reducing the availability of prey for predators.
Therefore, when there is an internal equilibrium point of the system (5.2.1), there will also be an axial equilibrium point (1−qE,0).
Local Stability Analysis
The consequence of this for (0,0) in the original x−y system is that a trajectory will only approach (0,0) when the rate of approach from x to 0 is faster than the rate of approach from y.
Global Stability Analysis
To determine an appropriate harvest effort that will ensure the conservation of both prey and predator populations, we need the equilibria (0,0) and (1−qE,0) to saddle and the internal equilibrium (¯x ,y) to be stable in nature. ¯. Thus, as long as the condition c < bm holds, the non-trivial equilibrium point for the system (5.2.1) is locally stable when it exists. Finally, we summarize all the conditions (for the three cases defined in Table 3.2) on effort E to achieve an environmentally sustainable harvest (for the system (5.2.1)) in Table 5.2.
Stability analysis gives us the way to determine an appropriate level of effort since a randomly chosen level of effort can lead to the extinction of the population.
Stability of the Modified Ratio-Dependent Model with Prey Harvesting
Existence of Equilibrium Points
From this relationship, we can conclude that whenever α >1, supplying extra food to predators reduces the pressure on.
Local Stability Analysis
Defining ∆ as an expression under the term of the square root ofxi, we obtain the conditions (see Table 5.3) for the model parameters, required for the existence of the above equilibrium points. However, (1−qE,0) always exists to ensure the existence of the internal equilibrium point. Note that, the condition for the saddle nature of (1−qE,0) is satisfied whenever any of the interior equilibrium points exists.
We study the stability of internal equilibrium points (for all three cases defined in Table 3.2) when (0,(b−α)ξ) either exists with saddle nature or does not exist at all.
Global Stability Analysis
Here it is clearly seen that ecologically sustainable harvesting is possible in all three cases, while without additional food is only possible in case 3. Choosing α > 1 results in an increase in the upper limit for ecologically sustainable harvesting (corresponding to the global stability of the internal equilibrium point) in compared to the case without additional food. As a result, by selecting the appropriate level of additional food supply, a higher level of effort can be achieved.
However, an appropriate level of additional food (for example, α = 2 and ξ = 0.5 for which coexistence is achieved for 0 < E < 0.633) ensures the stability of the internal equilibrium point and thus conservation of both species.
Optimal Harvesting Policy for the Ratio-Dependent Model
Since H is linear in control E, then the optimal control is combination of bang-bang and singular control [40]. Ee is also called singular control of the system (5.2.1) corresponding to singular solution (x,e y).e. According to the optimal harvest policy, it follows that if the size of the prey population is at some point greater than the optimal equilibrium prey population size ex = 0.1358, i.e. x(t) > 0.1358, then the harvesting effort is chosen to be Emax = 0.2667 until the prey size is reduced to ex.
The optimal trajectory is a combination of two trajectories, one (dashed) corresponding to the maximum harvesting effort Emax = 0.2667 while the second one corresponds to the single effort Ee = 0.1308.
Optimal Harvesting Policy for the Modified Ratio-Dependent Model
This can be ensured as long as the cost per unit amount of biomass of additional food satisfies. In addition, in this case there is an additional financial burden due to the supply of additional food. For this purpose, we refer to the example of gathering prey without an additional feeder, as stated in the previous section.
In this case, the harvesting effort may even be greater than the corresponding harvesting effort for the system without additional food.
Conclusion
We begin by regressing the modified ratio-dependent predator-prey model (2.1.3) with additional food (non-prey) provided to the predators given by , . Here the parameter α characterizes the quality of the additional food, as it is inversely proportional to the nutritional value of additional food. In other words, as the quality of additional food deteriorates (improves), the prey population closes (declines).
This clearly indicates that the quality of additional food has a direct role in the control of prey animals (pests).
Minimum Time Optimal Control Problem
The singular optimal solution is therefore a point (ˆx,y) in the state space, where ˆˆ x is the positive solution of the equation.
Switching Regions
From the above observation, we conclude that the transition of the control variable α from αmax to αmin (αmin to αmax) at time τ can occur only in the regions R1 and R3 (R2 and R4), where the regions are given by,.
Numerical Illustration and Discussion
Usually, the system (2.1.3) with the initial state P1 evolves under the control α=αmin until the solution trajectory intersects the switching curve P Q1. At the point of intersection on the curve P Q1, the rail now develops under control α=αmax until it intersects the second switching curve P Q2. At the intersection point at P Q2, the control switches again to α = αmin until the trajectory reaches its target state.
At the intersection of the transition curve, the applied control changes to α = αmin until the trajectory reaches its target state.
Conclusion
Chattopadhyay, Predator-prey ratio dependent model: the effect of environmental fluctuation and stability, Nonlinearity, vol. Beretta, Global qualitative analysis of a predator-prey ratio-dependent system, Journal of Mathematical Biology, vol. Arditi, On deterministic extinction in models dependent on the predator-prey ratio, Bulletin of Mathematical Biology, vol.
Petrovskii, Self-organized spatial patterns and chaos in a ratio-dependent predator-prey system, Theoretical Ecology, vol. Sheen, Turing instability for a ratio-dependent predator-prey model with diffusion, Applied Mathematics and Computation, vol. Haque, Ratio-dependent predator-prey models of interacting populations, Bulletin of Mathematical Biology, vol.
