Stochastic time-optimal control and sensitivity studies for additional food provided prey-predator systems involving Holling type-IV functional response

In this study we consider an additional food provided prey-predator model exhibiting Holling type-IV functional response incorporating the combined effects of both the continuous white noise and discontinuous Lévy noise. We prove the existence and uniqueness of global positive solutions for the proposed model. We perform the stochastic sensitivity analysis for each of the parameters in a chosen range. Later we do the time optimal control studies with respect quality and quantity of additional food as control variables. Making use of the arrow condition of the sufficient stochastic maximum principle, we characterize the optimal quality of additional food and optimal quantity of additional food. We then perform the sensitivity of these control variables with respect to each of the model parameters. Numerical results are given to illustrate the theoretical findings with applications in biological conservation and pest management. At the end we briefly study the influence of the noise on the dynamics of the model.


Introduction
The complex natural ecosystems present around us kindled great attention of many ecologists and mathematicians to the mathematical modeling of ecological systems in the last few decades.The interaction among species in these ecosystems can be of several forms like competition, mutual interference, prey-predator interactions and so on.The very first ecological models are framed from the pioneering works of Alfred J.
Lotka [1] and Vito Volterra [2] in 1925.Various complex models are framed and studied ever since.
The basic component of prey-predator systems is a functional response, which is defined as the rate at which each predator captures prey [3].A few examples of functional responses include Holling functional responses [4], Beddington-DeAngelis functional responses [5], Arditi-Ginzburg functional responses [6], Hassell-Varley functional responses [7] and Crowley & Martin functional response [8].In this work, we study the models exhibiting Holling type IV functional response.Some of the organisms that display Holling type IV functional response in nature are found in [9,10].In this functional response, the predator's per capita rate of predation decreases at sufficiently high prey density, due to either prey interference or prey toxicity.
In recent decades, many pioneering dynamical modeling works [11][12][13][14][15][16] reveal that the provision of additional food to predators plays a vital role in controlling the dynamics of the system.Recently, the authors in [14] have studied additional food provided deterministic prey-predator systems involving Holling type IV functional response.Also, the authors in [16,17] studied the optimal control problems of deterministic prey-predator systems involving Holling type IV functional response with the quality of additional food and the quantity of additional food as the control parameters respectively.Often it is observed that the parameters in an ecosystem are effected by the environmental fluctuations [18].Therefore in recent years, many researchers have drawn their attention to stochastic models.
Most stochastic prey-predator models are driven by the Brownian motion, which captures the continuous noise.Authors in [19,20] studied the deterministic and stochastic dynamics of a modified Leslie-Gower prey-predator system with simplified Holling type IV functional response.However, the sudden changes in environment like toxic pollutants, floods, earthquakes and so on, cannot be captured by the Brownian motion as it is a continuous noise.Hence, addition of a discontinuous noise, like Lévy noise, to the prey-predator system with Brownian motion makes the models more realistic.[21] uses the stochastic averaging method to analyze the modified stochastic Lotka-Volterra models under combined Gaussian and Poisson noise.[22] studies the dynamics and dynamics of a Stochastic One-Predator-Two-Prey time delay system with jumps.
To the best of our knowledge, there is no study of additional food provided stochastic prey-predator system with jumps.Secondly, the optimal control studies of Stochastic Differential Equations with Jumps (SDEJ) were not performed on prey-predator systems.Lastly, very few works involved Holling type-IV response which incorporates the most important group defence property.Motivated by these observations, in this work, we study the optimal control problems for additional food provided stochastic Holling type IV prey-predator systems under combined Gaussian and Lévy noise.
The article is structured as follows: Section 2 introduces the stochastic prey-predator model with Holling type-IV functional response and additional food with intra-specific competition among predators.The existence of global positive solution for this model is briefly discussed in Section 3. The time-optimal control problem is formulated and the optimal quality and quantity of additional food is characterized in Section 4.
Section 5 illustrates the key findings of the analysis through numerical simulations in the context of both biological conservation and pest management.Finally, Section 6 presents the discussions and conclusions.

The Stochastic Model
Let N and P denote the biomass of prey and predator population densities respectively.In the absence of predator, the prey growth is modelled using logistic equation.Further, we assume that the prey species exhibit Holling type-IV functional response towards predators.We also assume that the predators are supplemented with an additional food of biomass A, which is uniformly distributed in the habitat.Incorporating these assumptions, the prey-predator dynamics with Holling type-IV functional response along with additional food for predators can be described as: For a complete analysis of model (1), the reader is referred to Vamsi et.al., [15].
In addition, we also assume that the predators exhibit intra-specific competition.We capture this competition in similar lines with [23,24].Accordingly, the system (1) gets transformed to the following system.
The biological descriptions of the various parameters involved in the systems (1) and ( 2 In order to reduce the complexity of the model, we non-dimensionalize the system (2) using the following non-dimensional parameters.
. Accordingly, system (2) gets reduced to the following system.
In real world scenarios, environmental fluctuations affect the dynamics of the system.In order to capture these fluctuations, we introduce the multiplicative white noise terms into (3).As in [23,25,26], we now suppose that the intrinsic growth rate of prey and the death rate of predator are mainly affected by environmental noise such that where W i (t) (i = 1, 2) are the mutually independent standard Brownian motions with W i (0) = 0 and σ 1 and σ 2 are positive constants and they represent the intensities of the white noise.Also, the system can go through huge, occasionally catastrophic disturbances.Since white noise is a continuous noise, it cannot capture sudden environmental changes.To cater to these, we also apply a discontinuous stochastic process as Lévy jumps to model these abrupt natural phenomenon as in [22,27].
We now perturb r and m 1 with discontinuous Lévy noise in addition to the continuous white noise.So, we have According to the Lévy decomposition theorem [28], we have N (t, dv) = N (t, dv) − λ(dv)t, where N (t, dv) is a compensated Poisson process and N is a Poisson counting measure with characteristic measure λ on a measurable subset Y of (0, +∞) with λ(Y) < ∞.The distribution of Lévy jumps L i (t) can be completely parameterized by (a i , σ i , λ) and satisfies the property of infinite divisibility.Now, by incorporating noise induced parameters (4) into the reduced deterministic system of equations (3), we get the following additional food provided stochastic prey-predator system exhibiting Holling type-IV functional response along with the environmental fluctuations captured using the white noise and Lévy noise.

Existence of Global Positive Solution
In order to study the stochastic dynamics of (5), we first prove that the system (5) has a unique global positive solution.
The above theorem for existence of solutions of ( 5) can be proved in similar lines to the proof in [22] using the Lyapunov method.

Stochastic Time-Optimal Control Problems
In this section, we formulate and study the stochastic time-optimal control problems for prey-predator systems involving Holling type-IV functional response where the predator is provided with additional food.

Quality of additional food as control parameter
In this subsection, we characterize the optimal quality of additional food for driving the system (5) to a desired equilibrium state in minimum time using the stochastic maximum principle.So, we fix the quantity of additional food ξ > 0 to be a constant and choose the objective functional to be minimized for this stochastic time optimal control problem as follows.
From the Sufficient Stochastic Maximum Principle [29] for the optimal control problems of jump diffusion, we characterize the optimal solution of the stochastic time optimal control problem with state space as the solutions of ( 5) and the objective functional (6).
Let (p * , q * , r * ) be a solution of the adjoint equation in the unknown processes p(t) ∈ R 2 , q(t) ∈ R 2×2 , r(t, z) ∈ R 2 satisfying the backward differential equations The Hamiltonian associated with this control problem is defined as follows.
H(t, x, y, α, p, q, r) From the Arrow condition in the sufficient stochastic maximum Principle [29], we have Hence the optimal control α * should satisfy the following condition.
Since the analytical solution of ( 7) is complex to solve, we numerically simulate these results in section 5.

Quantity of additional food as control parameter
In this subsection, we characterize the optimal quantity of additional food for driving the system (5) to a desired equilibrium state in minimum time using the stochastic maximum principle.So, we fix the quality of additional food α > 0 to be a constant and choose the objective functional to be minimized for this stochastic time optimal control problem as follows.
From the Sufficient Stochastic Maximum Principle [29] for the optimal control problems of jump diffusion, we characterize the optimal solution of the stochastic time optimal control problem with state space as the solutions of ( 5) and the objective functional (10).
Let (p * , q * , r * ) be a solution of the adjoint equation in the unknown processes p(t) ∈ R 2 , q(t) ∈ R 2×2 , r(t, z) ∈ R 2 satisfying the backward differential equations The Hamiltonian associated with this control problem is defined as follows.
From the Arrow condition in the sufficient stochastic maximum Principle [29], we have Hence the optimal control ξ * should satisfy the following condition.
Since the analytical solution of ( 7) is complex to solve, we numerically simulate these results in section 5.

Existence and Uniqueness of Solutions for the Forward Backward Stochastic
Differential Equations with Jumps (FBSDEJ) We so far obtained the adjoint equations ( 7), (11) for the state equations ( 5) and the objective functional ( 6), (10) using the sufficient stochastic maximum principle respectively.Upon simplifying the results obtained from the arrow condition ( 9), ( 13) from earlier two subsections, we see that the optimal controls are given by In this section, we now prove the existence of optimal controls by proving the existence of the solutions for the FBSDEJ (( 5),( 7),( 11)) which establishes the existence of (x * , y * , p * 1 , p * 2 ) for all simulation purposes.Using the theorem in [30], we now prove the existence of the optimal controls (14) in the following theorem.
Proof.Let (X t ) t≥0 be the solution of the Stochastic Differential Equation with Jumps(SDEJ) Here the term b(X t ) denotes the drift coefficient, the term σ(X t ) denotes the diffusion coefficient and the term Γ(v) denotes the poisson term coefficient.
The theorem 1 in section 3 guaranties the monotonicity and Lipschitz continuity of the drift coefficient, the diffusion coefficient and the poisson term coefficient of the state equations (5).
Following the the existence and uniqueness theorem of FBDSDEJ in [30], we are only left to prove the monotonicity and Lipschitz continuity of the drift and diffusion terms of the adjoint system of equations (7).From (7), due to the positivity of state variables guaranteed by theorem 1, the drift term and the diffusion terms are given as follows.

b(X
Since the drift coefficient is a linear combination of adjoint terms (p 1 , p 2 ), the monotonicity and Lipschitz continuity are guaranteed.
In addition to this, the diffusion coefficient is independent of the adjoint terms (p 1 , p 2 ).Therefore, the monotonicity and Lipschitz continuity are guarenteed for the diffusion coefficients.
A. Applications to Biological Conservation: The subplot (1a) depicts the optimal state trajectory of the system (5) from the initial state (2, 8) that stabilizes over time around the state (16,90).The subplot (1c) gives the phase diagram which shows the trajectories are stabilized over high values of prey and predator.The subplots (1d) and (1e) depicts the optimal quality and quantity of additional food respectively.
These plots show that the high quality of additional food is required to achieve biological conservation.Even if the quantity of additional food is lower, still we will be able to achieve biological conservation with higher quality of additional food.

B. Applications to Pest Management:
The subplot (2a) depicts the optimal state trajectory of the system (5) from the initial state (2, 8) that reaches the nearly prey-elimination stage around the state (5,90).The subplot (2c) depicts this property more clearly through the phase diagram where it reaches the lowest prey value over the time.The subplots (2d) and (2e) depicts that a lesser quality of additional food and a lower quantity of additional food is good enough to achieve pest management where pest is viewed as prey.

Figure 1 :
Figure 1: This figure depicts the simulations of time-optimal control problem with respect to the control parameters in the context of biological conservation.

Figure 2 :
Figure 2: This figure depicts the simulations of time-optimal control problem with respect to the control parameters in the context of pest management.

Table 1 :
Description of variables and parameters present in the systems (1), ) are described in Table1.