An Optimal Control Study with Quantity of Additional food as Control in Prey-Predator Systems involving Inhibitory Effect


 Additional food provided prey-predator systems have become a significant and important area of study for both theoretical and experimental ecologists. This is mainly because provision of additional food to the predator in the prey-predator systems has proven to facilitate wildlife conservation as well as reduction of pesticides in agriculture. Further, the mathematical modeling and analysis of these systems provide the eco-manager with various strategies that can be implemented on field to achieve the desired objectives. The outcomes of many theoretical and mathematical studies of such additional food systems have shown that the quality and quantity of additional food play a crucial role in driving the system to the desired state. However, one of the limitations of these studies is that they are asymptotic in nature, where the desired state is reached eventually with time. To overcome these limitations, we present a time optimal control study for an additional food provided prey-predator system involving inhibitory effect with quantity of additional food as the control parameter with the objective of reaching the desired state in finite (minimum) time. The results show that the optimal solution is a bang-bang control with a possibility of multiple switches. Numerical examples illustrate the theoretical findings. These results can be applied to both biological conservation and pest eradication.


Introduction
Prey-predator systems where predators are provided with additional food supplements are being extensively studied by biologists, ecologists (both theoretical and experimental ecologists) as well as mathematicians [16,31,33,36,46,48]. This is mainly because the method of providing additional food not only e ectively aids the conservation of threatened species [16,30,31] and but also facilitates the control of harmful and invasive species [23,47,48]. Nonetheless, adequate care needs to be taken while providing additional food because lack of proper vigilance could lead to adverse consequences [30,32]. These systems are generally modeled as di erential equations and their dynamics are studied to analyse the behaviour of such systems. Some of the ndings of mathematical studies and analysis of additional food provided prey-predator systems [11,12,13,24,25,27,28,34,35,39,41] reveal that the additional food plays a crucial role in the dynamics of these systems, a ecting the eventual state and stability of species. These ndings are in line with the experimental observations when additional food is provided to the predators [5,4,16,48] In any mathematical model that describes a prey-predator interaction, one salient feature that characterizes the nature of this interaction is the Functional Response, which is de ned as the rate at which prey is captured by the predator [17]. One of the many functional responses displayed by di erent predator species is the Type IV functional response where the predator's catchability reduces when prey density is high. This could be a consequence of prey interference or toxicity. In other words, the prey tend to exhibit group defense when in large densities as a survival mechanism [7,10,49]. The type IV functional response is also called the functional response involving inhibitory e ect. For example, caterpillar species Malacosoma disstria display group defense against their natural enemy parasitoids to reduce risk of predation [22]. As a result of forming bigger groups, the parasitoid does not succeed in catching its prey in spite of multiple attacks.
Recently, in [40] and [45], the authors have modeled and studied the following additional food system involving type IV functional responsė The parameter α denotes the e ciency of predator to convert the additional food available into predator biomass relative to conversion of prey. α is inversely proportional to the nutritional value of the additional food and directly proportional to the handling time of the additional food. In this study, we use the parameter α to represent the Quality of additional food provided to the predators. Additional food is termed high quality if α < β δ and low quality otherwise. The parameter η = e e ϵ ϵ denotes the ratio of search time of predator per unit food item of additional food and prey relative to nutritional values of additional food and prey. The term ηA = e e (A)/ ϵ ϵ represents the quantity of additional food discernible to the predator with respect to the prey relative to the nutritional value of prey to the additional food. Let N = ax; t = rT; P = y ra c ; which gives dN = adx; dt r = dT; dP = ra c dy. Now, using the following transformations γ = K a , ω = ba , β = e r , δ = m r , and taking ξ = η A a , we get the corresponding non-dimensionalized system as follows: In the above non-dimensionalised system, γ and ω are parameters that represent the carrying capacity and inhibitory e ect respectively. In this study, we consider the parameter ξ to denote the Quantity of additional food provided to the predators. Let f (x) = x ( +αξ )(ωx + )+x and g(x) = − x γ (( + αξ )(ωx + ) + x). Then, the system (1.3) -(1.4) can be modi ed as follows: The outcomes of the works [40] and [45] concede the importance of quality and quantity of additional food that is provided to the predator reinforcing the inferences from the studies [18,19,30] and [50]. However, in [40] the desired states are shown to be reached eventually with time as asymptotes. This makes the outcomes not that practically helpful. To overcome this limitation, we propose to study the system (1.3) -(1.4) further in the direction of achieving controllability in minimum ( nite) time. Also, we see from various ecological studies that the quantity of additional food plays a signi cant role in the dynamics of the system [4,44,47,48,50]. Since the nutritional value and the conversion factor are always xed for a given food item, the quantity of consumption thus plays a major role in determining the e ect of providing the additional food. Also, it is practically easier to vary the quantity of food supplements provided than varying the quality. Therefore, the study of controllability using the quantity of additional food becomes relevant.
Motivated by the above discussion, in this article, we study the role of quantity of additional food ξ in in uencing the global dynamics of the additional food provided system (1.3) -(1.4) in achieving the desired state in minimum ( nite) time. To that end, we rst determine the possible admissible states that could be reached by varying the quantity in the range [ξ min , ξmax]. We then formulate and study a Time optimal control problem using quantity of additional food as the control parameter. We use Filippov's Existence Theorem to prove the existence of optimal solution and Pontryagin's Maximum Principle to obtain the characteristics of the optimal solutions. Throughout the study, we keep the quality of additional food α xed. The outcomes of this work can bene t eco-managers in providing additional food to species depending on the objective being biological conservation or pest management. Optimal control studies for additional food provided type II functional response have been done in the works [37,38]. Since type IV response is a more general form of type II response, it becomes signi cant to perform control studies on systems involving type IV response. The controllability studies with respect to quality of additional food keeping the quantity xed has its own relevance and is under progress [1].
The section-wise division of this article is as follows: In the next section, we discuss the biological relevance of having quantity as the control parameter. In section 3, we study the role of quantity of additional food and its in uence on the dynamics of the type IV additional food provided system and determine the possible admissible equilibria. In section 4, we formulate and study the time optimal control problem with quantity of additional food as the control parameter following which in section 5, we discuss optimal solution strategies and applications to pest management. Then, in section 6, we illustrate the theoretical results using numerical examples followed by a section on the role of inhibitory e ect. Finally, we present the discussion and conclusions in section 8.

Biological Relevance of Quantity as a Control Parameter
The ndings of several ecological and entomological studies supported by experimental observations show that the quantity of additional food provided to the predator species impact their survival, oviposition rate, longevity, fecundity as well as predation rate [2,4,14,29,44,47,48,50,51]. This is more so applicable in the context of bio-control or pest management where natural enemies of these invasive species or pests are mass reared for attacking their target prey as a part of habit management [20]. Plutella xylostella (diamondback moth) is a herbivore whose parasitoid is Diadegma semiclausum. The parasitoid D. semiclausum was tested with nine sugars and honey dew sources to provide carbohydrate rich food in its adult stages [50]. The gustatory response of D. semiclausum was found to be high when their concentration was 1M and the response started to drop with decreasing concentration.
Macrolophus pygmaeus is an omnivorous mirid predator which is one of the widely used natural enemies in augmentative biological control against pests like white ies, arthropod pests, tomato pests etc., in Europe [29]. Ephestia kuehniella eggs are a factitious food which are used as an alternate source to rear these species. The results from the experimental studies [47] reveal that as the number of E. kuehniella eggs provided per three days increase from 10 to 40, the M. pygmaeus species reduced their development time signi cantly, females had better oocyte counts and nymphs increased their survival. However, it cannot be concluded that high quantity of additional food supplements is always bene cial. For instance, Nesidiocoris tenuis, which is a zoophytophagus natural enemy of tomato pest Tuta absoluta, is provided with sugars of concentration 0.5M and 1M, it is found that the immature furvival of N tenuis was lower when provided with 1M sucrose than with 0.5M sucrose [43].
Thus, the in uence of quantity of additional food supplements on predator species (and hence on the ecosystem) depends on the quality of additional food [48]. Since the quality of additional food de ned above not just implies the calori c value of the item but the handling time and the relative e ciency, the predators response to a speci c item also plays a major role in determining the quality of additional food which can be treated constant for a speci c item. Thus, using quantity of additional food to study the controllability of the system very pertinent to understand the dynamics of the system and also to drive the system to a certain desired state in minimum time.

Role of Quantity in the Global Dynamics of the Additional food system
In this section, we will outline the global dynamics of the additional food system (1.3) -(1.4) and the role of quantity of additional food in in uencing the eventual state and stability of the system. The type IV additional food provided system (1.3) -(1.4) admits four equilibrium points: the trivial equilibrium E * = ( , ), the axial equilibrium point E * = (γ, ), and two interior equilibria E * = (x * (α, ξ ), y * (α, ξ )) and E * = (x * (α, ξ ), y * (α, ξ )) given by , g(x * ) .
The interior equilibria exist only when the term under the square root (β − δ) − ω[δ( + αξ ) − βξ ] > and x * < γ and x * < γ. The analysis carried out in the works [40] and [45] show that the dynamics of the system (1.3) -(1.4) depend on the dynamics of the initial system (without additional food) and nature of its isoclines. Also, for maintaining brevity, we too shall con ne this work to Condition I of the initial system as considered in [40], where interior equilibrium is not admitted by the initial system. In other words, we consider the following condition holding true: Under this condition, we get three di erent cases for the additional food system. The gure 1 depicts the di erent types of the prey isocline that occur under the condition -I.  The phase space analysis of the additional food system reveals that along with the nature of the prey isocline curve (as shown in gure 1), the eventual state of the system and stability of the equilibria also depends on the values assumed by the two parameters α and ξ with respect to the three bifurcation curves and the discriminant curve given below:  In the previous sections, we saw that the quality of additional food can be classi ed into high quality and low quality depending on the position of the parameter α with respect to β δ . From the discussion above, we see that the additional food system does not admit any interior equilibria when the additional food is provided of low quality under the Condition -I. This is due to the dynamics established in [40] which states that: If the type IV system does not admit interior equilibrium in the absence of additional food, then the additional food system shall never admit interior equilibria by providing low quality additional food α > β δ . Thus, in this work, we will discuss only the provision of high quality additional food when the system has no interior equilibrium in the absence of additional food. Interior Equilibria: The terms P, Q, R and S in the table are obtained by solving the discriminant curve and the three bifurcation equations for ξ .
From the table -2, we see that by providing appropriate quantity of high quality additional food, the system can be driven to any desired state. We will now determine the admissible states that can be reached in nite time when additional food of high quality is provided and by varying the quantity in the range [ξ min , ξmax], where ξ min (ξmax) represents the lowest (highest) quantity of additional food provided. We observe from the table -2 that in the context of pest eradication or pest management, where the terminal state eliminates prey (or maintains prey density with least harm to the system), we need to have ξ > β β−δα , so that the trajectories which emerge from below the prey isocline curve touch the y-axis in nite time [37,40] (or reach the desired state su ciently close to x = ). Otherwise, there is no possibility of any trajectory that moves towards the predator axis (y-axis). In some cases of bio-control it is also preferred that the prey (pest) are not completely eliminated from the system because predator tends to damage the crops in the ecosystem after the target prey gets eliminated [6,43]. In such cases, prey are maintained at low densities such that they do not damage crops. Based on the discussion above, we will now state a result in the context of pest management: Proof.
From the dynamics presented in the table -2, we observe that when β − δα > and ξ > δ β−δα , the trivial equilibrium is unstable and the axial equilibrium (γ, ) is a saddle whose unstable manifold moves towards origin on the prey axis.
Thus, for any solution initiated under the prey isocline curve, the saddle of axial equilibrium pushes the trajectory towards origin whose unstable nature in turn drives the system asymptotically towards the predator axis eventually eliminating the prey. On the other hand, for the solutions generating above the prey isocline curve, the prey isocline curve itself acts as an unstable manifold of the axial equilibrium thereby driving the state towards prey elimination. Let us now consider the case of co-existence of species in the additional food system. With that goal, we must drive the additional food system towards interior equilibrium point which would be admissible. Since we see from the dynamics of the system (1.3) -(1.4) given in table -1 that second interior equilibrium is a saddle throughout its existence, we do not wish to drive the system to that equilibrium point. However, from the table 2, we see that under certain conditions, there is a possibility of occurence of a homoclinic orbit especially when the interior equilibrium E * is unstable. Under those conditions, it may not be possible to reach E * from any initial point. Thus, we will consider E * to be admissible only when it is unique, i.e., when the prey component of E * = (x , y ) is such that x > γ. Henceforth, we will refer to E * = (x * (ξ ), y * (ξ )) as the interior equilibrium. Let us consider the interior equilibrium E * = (x * (ξ ), y * (ξ )) for a xed ξ > . The prey and predator component are given by Now, solving (3.1) for ξ , we get The above equation (3.4) gives us the set of all admissible equilibrium points for the additional food system (1.3) -(1.4). This curve (3.4) intersects the y * -axis at y * = β β−δα . In order to understand the nature of the curve in (3.4), we consider the rst derivative of y * (ξ ) with respect to x * (ξ ): By obtaining the roots of the quadratic equation − ω(x * (ξ )) + ( ω + α − )x * (ξ ) + γ( − α) − = , we get the critical points of the admissible curve of all equilibria and based on that we can determine the nature of admissible points. The roots of the above quadratic equation are given by Depending on the existence of the above roots we get the following four cases of the nature of the admissible curve Depending on the nature of the admissible curve (3.4), for a chosen predator density, we may have one or more prey density options to drive the system to, depending on the quantity of additional food provided. We summarize them in the following result:

Proposition 2. For Prey Species Conservation:
Let the equilibrium E * be the unique interior equilibrium for the system (1.3) - (1.4). Then, for < x < γ, there exists a unique ξ such that (x, y(ξ )) is an admissible interior equilibrium for the system (1.3) -(1.4).

For Predator Species Conservation:
Let the equilibrium E * be the unique interior equilibrium for the system (1.3) - (1.4). Let A = β β−δα and let xc denote the critical point in Case III with the corresponding predator component ymax. Let x C and x C be the two positive critical points in Case IV whose predator components are denoted by y min and ymax respectively.

Time Optimal Control Studies
We will now formulate and study a time optimal control problem for the additional food provided system (1.3) -(1.4) that would drive the system from a given initial state (x , y ) to the desired terminal state (x,ȳ) in minimum time using optimal quantity of additional food provided to the predator species, with for a xed quality of additional food. Let α > be xed, and let the parameter representing the quantity of additional food be the control function ξ (t) whose range is in the interval [ξ min , ξmax]. Then the time optimal control problem (a Mayer Problem of Optimal Control [8]) can be de ned for the system (1.3) -(1.4) as follows: Using the equations of the system (1.5) -(1.6), we get the relation The control problem 4.1 is of the form of a Mayer time optimal control problem (Appendix -A) with n = , m = and The set A for this problem (4.1) is the subset of the tx -space (R + ), i.e., A ⊂ R + which is nothing but the solution space. We de ne the set of all admissible solutions to the above problem is given by ξmax] .

Existence of Solution for the Optimal Control Problem
We wish to obtain a solution from the set Ω which minimizes the time to reach the terminal state (x(T), y(T)), that, in turn becomes the optimal solution for (4.1). We will now establish this in the following theorem by proving the existence of an optimal control using Filippov's Existence Theorem (refer to Appendix -A) which drives the system to a desired terminal state in minimum time.

Theorem 1. The optimal control problem 4.1 admits an optimal control ξ (t) provided the terminal state is admissible (Propositions 1 and 2) and the the set of admissible solutions Ω is non-empty.
Proof. In order to prove this result, we will show that all the following conditions of the Filippov's Existence Theorem are satis ed by the considered control problem: 1. The set A is compact.
We will now show that the optimal control problem (4.1) satis es all the properties above.
(i) Whenever βξ −δ δξ < , we know from the global dynamics of the system (Table -2) that the solution trajectories of the system (1.3 ) -(1.4) reach y − axis in nite time and are therefore closed and bounded. On the other hand, whenever βξ −δ δξ > , we have from the positivity and boundedness theorem [45] that the solutions are closed and bounded. Thus, we can conclude that A is compact and this proves condition 1.
Using the fact that z = f (x, y, ξ ) = β x+ξ (ωx + ) x+( +αξ )(ωx + ) y − δy, and from (4.2), we get Now, from (4.2), we see that and from (4.3) we have Now, substituting for ξ in the above expression (4.5) from (4.4) and simplifying the expression, we get Finally, after rearranging the above expression, we get The linear relation seen above (4.7) between z and z implies that the set Q(x, y) are convex segments. This proves the condition 4.
Thus, if the set of admissible solutions to the control problem (4.1) is non-empty, i.e., Ω ≠ ϕ, then the existence of absolute minimum can be guaranteed.

. Characteristics of the Optimal Control Solution
In this section, we will obtain the characteristics of the optimal control solution to the control problem (4.1) assuming that the optimal control solution exists. We do this using the necessary conditions for optimal solutions given by the Pontryagin's Maximum Principle [21].
First we de ne the Hamiltonian function associated with the optimal control problem (4.1) as H(x, y, ξ , λ, µ) := λ dx dt + µ dy dt where, λ and µ are co-state variables or adjoint variables. We expand the above expression using the system equations (1.3) -(1.4) and we get the Hamiltonian as By rearranging the terms, the above expression becomes Also, using the symbolically modi ed additional food system (1.5) -(1.6) based on the de nitions of the functions f (x, α, ξ ) and g(x, α, ξ ), we get the Hamiltonian to be The maximum principle states that if optimal solution exists, then the co-state variables satisfy a system of equations called the Canonical equations, also called the Adjoint system, given by . Using (4.10), the adjoint system can be expressed as Now, we di erentiate the Hamiltonian function (4.9) with respect to the control parameter ξ in order to obtain the characteristics of the optimal control.
We can observe that the optimal control ξ * (t) cannot be explicitly obtained from the above relation. Thus, when we di erentiate the Hamiltonian function further with respect to ξ , we get We can now conclude from the above expression (4.13) that the Hamiltonian function is a monotone with respect to the parameter ξ provided the we have ∂H ∂ξ ≠ . Using the Hamiltonian maximization condition of the maximum principle (which becomes a minimization condition in our case by the de nition of the objective function) [21], we see that H(x * (t), y * (t), ξ * (t), λ * (t), µ * (t)) ≤ H(x * (t), y * (t), ξ , λ * (t), µ * (t)) (4.14) ∀ξ ∈ [ξ min , ξmax] and ∀t ∈ [ , T]. Also, since the formulated control problem (4.1) is a time optimal control problem, the Hamiltonian function would turn out being a constant along the optimal trajectory and in particular, it assumes value -1 [9]. Hence, Thus, using equation (4.9), the condition (4.14), and the monotonicity property of Hamiltonian function with respect to ξ , we conclude that the optimal control solution ξ * (t) could be of bang-bang type if no singularity exists in any sub-interval of , T. This implies that optimal control function would assume the form: As mentioned above, singularity in optimal solution occurs when ∂H ∂ξ = for a sub-interval of [ , T]. In this situation, the optimal control function can no longer be obtained using the Hamiltonian minimization condition and the monotonicity property. To show that the optimal solution is of bang-bang type only, we must rule out the existence of a singular arc in any interval (t , t ), ⊆ [ , T]. Thus, to precisely the characterize the optimal control, we rst assume that singular solution exists and then show that singular arc actually does not exist.
Let ∂H ∂ξ = for some time instant t ∈ [ , T]. This implies that Rearranging the terms in the above equation, we get From the above equation we can infer that along the singular solution, the co-state variables behave as follows: (i) λ and µ will be of opposite signs when ωx − (α − )x + > . (ii) λ and µ will be of same sign when ωx − (α − )x + < . (iii) λ = and µ will be arbitrary when ωx − (α − )x + = The behaviour of the co-state variables along the singular solution shows that both of them cannot become zero simultaneously because that would lead to a contradiction with respect to the Hamiltonian along the optimal trajectory given by equation (4.15). Thus, to characterize the optimal trajectory along the singular solution, we di erentiate the equation (4.11) with respect to time. This gives us d dt Expanding the terms in the above expression using the additional food system (1.5) -(1.6), the adjoint system (4.11) and simplifying the terms using the condition (4.16) which is satis ed along the singular solution, the equation ( The above equation implies that along the singular solution, we have The above relation implies that if singularity occurs in the optimal solution of the control problem (4.1), then it does at the roots of the above cubic equation (provided the roots are real and positive). To get more insight into the points of singularity, we di erentiate equation (4.21) again with respect to time along singular solution assuming that at least one root of the cubic equation (4.22) is real and positive (denoted byx). Then, using equation (4.16) we get Thus, we conclude that if singularity occurs, it will be at points (x,ŷ) and there is no possibility for a singular arc. Hence, we note that that the optimal control can be established using the Hamiltonian minimization condition and monotonicity property as guessed above. Summarizing the above analysis we will now state a result that characterizes the optimal solution of the control problem (4.1)

Theorem 2. The optimal control strategy for the time optimal control problem (4.1) is a combination of bangbang controls only, with possibility of switches occurring at points in the optimal trajectory. The Optimal control is given by
Now, based on the theorem stated above, we state a corollary to the existence theorem (Theorem 1).

Corollary 1.
If there exists an admissible path connecting the initial state (x , y ) and the terminal state (x,ȳ) involving a combination of bang-bang controls, then the optimal control problem (4.1) has an optimal solution.
In the previous section, we saw that if the system admits both equilibria, then the desired terminal state may not be admissible (or cannot be reached) if a homoclinic orbit exists. However, now we see from (4.25) that the optimal control strategy is a bang-bang strategy that alternates between extremes. Also, since the existence of homoclinc orbits depends on the choice of ξ * for a constant α, we see that if for one of the extreme values ξ min or ξmax, the homoclinic orbit does not exist, then we can still drive the system to the desired terminal state. We will depict this scenario though a numerical simulations in the later section.

Optimal Solution Trajectories and Applications to Pest Management
In this, section, we study the nature of the optimal solution trajectories and approach paths based on the switch points (if singularity occurs) and analyse the optimal solution. We will speci cally see the applications of these results in the context of pest management. For that, we rst consider the following equation whose roots give the points of singularity in the optimal solution In order to know about the existence and nature of the singular points, we need to establish the existence of number of roots of the above equation (5.1). Using the Descarte's rule of signs and thereby understanding the co-e cients and the parameters involved, we conclude that the the cubic equation (5.1) can have either one, two or no real roots. Accordingly, the phase space can be divided into various regions as follows: 1. Case I: F(x) has no positive roots: In this case, we can divide the phase space into two regions as follows: 2. Case II: F(x) has one positive root denoted byx. In this case, we can divide the phase space into four regions as follows: 3. Case III: F(x) has two positive roots denoted byx andx. Let us assume without any loss in generality that x <x. In this case, we can divide the phase space into ve regions as follows: Now, we will state and prove a result which describes the nature of switching (if it occurs), depending on the regions that divide the phase plane.

Proposition 3. The optimal control ξ * (t) along the optimal trajectory switches from ξ min to ξmax (or ξmax to ξ min ) in Regions Ia, IIa, IId, IIIb and IIId (or in Regions Ib, IIb, IIc, IIIa, IIIc and IIIe) only.
Proof. We know from the results of the previous section that when a switch occurs along the optimal trajectory, then at that instant we have ∂H ∂ξ = . From (4.12), we get Let us denote t = τ as the time instant at which switch occurs. Then we get Since (4.15) holds along the optimal trajectory, we get H(x(τ), y(τ), ξ (τ), λ(τ), µ(τ)) = − Using the de nition of Hamiltonian from equation (4.9), we get Now, substituting (5.4) in (5.3) followed by some simpli cation of the terms, we get Since we arrived at the condition that µ ≠ along the optimal solution, multiplying both sides of the above equation by α µ(τ) and using the expression for λ(τ) µ(τ) along optimal solution, we get After rearranging the above expression, we get From the nature of the curve (3.4) and the de nition of various regions, we see that µ(τ) is positive (negative) in the regions Ia, IIa, IId, IIIa, IIIb and IIId (Ib, IIb, IIc, IIIa, IIIc, and IIIe).
Using the equations (4.19), (4.20) and using the fact that σ(τ) = , we get From the optimal control strategy (4.25), we see that when the control switches from ξmax to ξ min (or ξ min to ξmax) at t = τ, then ∂H ∂ξ accordingly increases from negative to positive (positive to negative). Thus, dσ dt > (< ) at t = τ for the switch ξmax to ξ min (or ξ min to ξmax). Using observations along with the equations (5.6) and (5.7) we can conclude that the switch ξmax to ξ min (ξ min to ξmax) can occur in Ia, IIa, IId, IIIa, IIIb and IIId (Ib, IIb, IIc, IIIa, IIIc, and IIIe) only.
The above result (Proposition 3) shows that for reaching an interior point in the solution space, the optimal control may involve switching between values (ξ min ) and (ξmax) at multiple points. This means that in order to achieve biological conservation, the additional food supply could involve switches in the amount of food that is supplied so that co-existence of species is achieved. Now, we will focus on the relevance of the mathematical analysis hitherto in the case of pest management.
From the existence theorem (Theorem 1), we see that in the context of pest management, in order to reach the desired state at nite time, our goal is to reach the terminal prey density as x(T) = . Now, considering the desired terminal state to be ( , y(T)), we state two important results that establish the characteristics of the co-state variables at the terminal time t = T and the characteristics of the optimal control throughout the optimal trajectory leading to pest eradication.
Proof. We will use the zero solution of the linear system of the co-state variables (4.11) along the optimal path to prove this theorem. The matrix form of the system (4.11) is given by From the expansion of the matrix co-e cients given above, we observe that • a (t) > • The sign of b (t) can be determined based on the sign of the term + αξ − ξ .
• If ξ (t) = ξmax, then b (t) > by the hypothesis of the theorem • a (t) can either be negative or positive given the values of the state variables and parameters The characteristic equation of the system (5.9) is given by m + (a (t) + b (t))m + (a (t)b (t) + a (t)b (t)) = (5.10) We know that based on the properties of the functions (a (t) + b (t)) and (a (t)b (t) + a (t)b (t)) used in the characteristic equation (5.10) we can understand the qualitative properties of the solution of the system (5.9). Assuming that λ(T) < , using the continuity of functions (a (t) + b (t)) and (a (t)b (t) + a (t)b (t)), we can imply that there exists a left neighbourhood of T in the interval [ , T], say [s, T], such that λ(t) < and µ(T) < for all t ∈ [s, T]. As a result, we have ∂H ∂ξ < and consequently ξ opt (t) = ξmax. The proof of this theorem would be complete if we can show that s = .
Now, using the qualitative behaviour of the zero solution of the system (5.9), we will show that the initial values for λ and µ can be chosen in such a way that λ(t) < and µ(t) < for all t ∈ [ , T] thereby proving this theorem. Let us consider two cases based on the sign of the function b (t). In this case, b (t) > and this fact is not su cient to conclude anything on the sign of the discriminant of the characteristic equation (5.10). Thus, there is a possibility that the solution of the system (5.9) that begins in the third quadrant of the λµ -space drifts to another quadrant with time. From the prey-predator isoclines causing the interior equilibrium, we can observe that at time t = T, for x(T) = , we have y(T) > + αξmax. This means that the zero solution of the system (5.9) behaves as a saddle when time t is closer to the terminal time T. Thus, in order to ensure that the solution of the system (5.9) remains in the third quadrant, the initial values must be chosen such that µ( ) is far away from zero on the negative µ -axis and λ < , so that when the co-state solution nears negative λ -axis as t → T, then the saddle nature of the zero solution prevents it from going out of the third quadrant in the λµ -space.
The initial value chosen for co-state variables is (λ( ), µ( )) = (− , − ). Based on the table -2, for this example S = . and accordingly S < ξmax. Based on our earlier analysis, the solution trajectories should lead to prey elimination. This example shows how the results obtained can be applied in case of pest management. This example illustrates Theorem 3 where the optimal control does not undergo any switch. Also, the co-state variables are negative throughout the optimal trajectory. This examples shows that when high quality additional food of maximum quantity is provided to the predators, then prey (pest) eradicated from the ecosystem. The desired terminal state is reached in T = . units of time.

Numerical Illustrations
In this section, we present numerical examples that illustrate the results established in the previous section. We have performed the simulations on MATLAB software. In order to obtain the optimal solution, rst we xed the initial and terminal states and accordingly the range of the control parameter ξ (t) as [ξ min , ξmax]. Using the fact that the Hamiltonian function is constant with value − along the optimal trajectory, we x the initial values of co-state vectors with various combinations of trial and error processes with the goal of reaching the terminal state. We then simulate the systems (1.3) -(1.4) and (4.11) using the 4th order Runge-Kutta routines and switched the control using the switching function ∂H ∂ξ while monitoring the Hamiltonian function until the terminal state was reached. We used the step size h = . for the simulations, based on which the time units obtained at the end of simulations are accordingly re-scaled by multiplying T × − . , ω = . , ξ min = . , and ξmax = . . The initial value chosen for co-state variables is (λ( ), µ( )) = (− , . ). Based on the table -2, for this example we have P = . , Q = .
and S = . and accordingly ξ min < P < Q < ξmax < S. We see here that this situation is an example of Case Ia where there exists an asymptotically stable limit cycle around the unstable interior equilibrium. The desired terminal state is reached in T = . units of time. This is an example where both the prey and predator densities are increased in the process of providing additional food of appropriate quantity. In particular, the predator density is significantly increased by predominantly providing additional food of high quantity. We observe that the optimal control switches twice depending on the switching function. This example depicts the case where multiple switches are involved to bring in co-existence of species illustrating the ndings Theorem 2. (frame E), whereas for ξ * (t) = ξ min , the trajectory drives the system to stable axial equilibrium leading to predator extinction (frame F). This example shows that additional food needs to be carefully provided because either of the extreme quantities is not desirable to the system. The optimal control trajectory is able to pass through the homoclinic orbit in order to achieve the desired outcome. The desired terminal state is reached in T = . units of time. We observe that the optimal control switches once depending on the switching function illustrating the ndings Theorem 2.

Figure 12:
This gure depicts the optimal trajectory of the time optimal control problem (4.1) from the initial state ( . , ) to the terminal state ( . , . ) with the parameters values γ = . β = . , δ = . , α = . , ω = , ξ min = . , and ξmax = . . The initial value chosen for co-state variables is (λ( ), µ( )) = ( , ). This is an example where the predator density is signi cantly brought down by providing low quantity of additional food initially and then increasing the quantity to sustain there and ensure co-existence. The desired terminal state is reached in T = . units of time. We observe that the optimal control switches once depending on the switching function illustrating the ndings of Theorem 2. Figure 13: This gure depicts the optimal trajectory of the time optimal control problem (4.1) from the initial state ( . , . ) to the terminal state ( . , . ) with the parameters values γ = . β = . , δ = . , α = . , ω = . , ξ min = . , and ξmax = . . The initial value chosen for co-state variables is (λ( ), µ( )) = (, ). This is an example where there is a homoclinic orbit present in the system when ξ = ξmax. However, in the optimal trajectory, we see that the system does not admit a homoclinic orbit for ξ = ξ min . Thus, we can drive the system to its desired terminal state optimally. For ξ = ξmax, we see that both the interior equilibria exist and in fact the initial point chosen for this example is the second equilibrium corresponding to ξmax which is a saddle leading to a homoclinic orbit that occurs for ξ < .
. Since ξmax = . in this case and ξ min = . for which the saddle equilibrium and therefore homoclinic orbit do not exist, we can drive the system to the desired terminal state by switching to ξ min .

Role of Inhibitory E ect
One of the distinguishing factors that make of the additional food system (1.3) -(1.4) is the type IV functional response. This is due to the inhibitory e ect or group defense exhibited by the prey when in large densities. In this section, we will describe some of the unique features and in particular, compare with the type II functional response.
Let x be the dimension-less prey density. Then the non-dimensionalised type IV functional response function is given by where the parameters α and ξ represent the quality and quantity of additional food respectively and ω represents the inhibitory e ect displayed by the prey. We see that when ω = , we get the functional response as which is nothing but the type II functional response in the presence of additional food [39]. Also, by com- Figure 14: This gure shows the type IV functional response curve with varying inhibitory e ect. The rst frame on the lest shows the case with ω = , which is the type II functional response. The second frame shows how the response reduces with increasing prey density as the parameter ω increases.
paring the expressions (7.1) and (7.2), we see that at low prey densities, both the functional responses behave in a similar manner. We can also observe from the gure 15 below that when ω = , i.e. when the response is of form type II, the rate of predation increases with increase in prey density and then reaches a saturation. Whereas for type IV response, after a certain prey density, the predation starts to reduce with increase in prey density. To understand this better, we di erentiate the functional response with respect to the prey density and taking df dx = , we get the critical prey density as From the above expression, we see that the critical prey density reduces with increase in inhibitory e ect and as a result, the functional response curve reaches its maximum for lower prey densities and then it starts to decline with increase in prey density. The high predation of predators at low prey densities is similar to the behaviour of the predators that exhibit type II response [39]. The parameter ω plays a crucial role in the nature of prey isocline, on which the dynamics of the system depend on. The prey isocline curve is given by Plotting the prey isocline curve ( gure -15) with changing inhibitory e ect parameter ω, we see that as ω increases, the curve moves upwards. This is indicative of the fact that when it intersects the predator isocline, the resultant equilibrium predator population increases with increase in inhibitory e ect of the prey. Also, in the absence of inhibitory e ect, i.e. for the type II system there is only one interior equilibrium where as for the type IV system, we have two interior equilibria which can exist. Therefore, by incorporating the inhibitory e ect of prey into the system with the parameter ω, we see that the system exhibits richer dynamics when compared to the type II system [40,45] .
Finally, even in the optimal strategy to achieve prey elimination, we see that the inhibitory e ect plays a crucial role in achieving the desired terminal state. The gure -16 depicts the optimal trajectories of the time optimal control problem (4.1) with same boundary conditions and parameters as for gure 9 with increasing inhibitory e ect. We have considered 6 di erent values of inhibitory e ect ω. We know that when ω = , the functional response is nothing but the Type II response. For same parameters with ω = , results in [38] shows that y(T) = . and T = .
. From the various cases presented above, we see that as inhibitory e ect increases, the natural enemy density required to eliminate the invasive species increases and the time required to achieve the desired outcome too increases. Figure 16: This gure depicts the optimal trajectories of the time optimal control problem (4.1) with same boundary conditions and parameters as for gure 9 with increasing inhibitory e ect.
We have considered 6 di erent values of inhibitory e ect ω. We know that when ω = , the functional response is nothing but the Type II response. For same parameters with ω = , results in [38] shows that y(T) = . and T = .
. From the various cases presented above, we see that as inhibitory e ect increases, the natural enemy density required to eliminate the invasive species increases and the time required to achieve the desired outcome too increases.

Discussion and Conclusions
Additional food provided prey-predator systems have been receiving enormous attention over the years by both the theoretical and experimental ecologists as well as mathematicians [3,15,31,36,46,45] since the outcomes of these studies are relevant in the biological conservation and bio-control of species in ecosystems. The role of quality and quality of additional food provided to predators and its impact on the ecosystem has also been discussed in various works [2,30,31,33,48].
Recently, in the works [40] and [45], the authors model and analyze an additional food provided preypredator system that involve type IV functional response. These studies reveal that by providing additional food of suitable quality and quantity to the predators, the system could be driven either to an interior equilibrium ensuring co-existence of species or to an axial equilibrium leading to prey/predator elimination. However, since the terminal states are reached as asymptotes, the practical application of these strategies become di cult. Several ecological studies and experimental observations demonstrate the crucial role played by the quantity of additional food in determining the state and stability of the system [2,4,14,29,47,50,51].
Thus, in this work, to overcome the limitation of asymptotes, we have studied a time optimal control problem to achieve controllability of the system in minimum ( nite) time with respect to quantity of additional food. First, we determined the admissible equilibria and the corresponding control necessary to maintain the system at that state. Then, we formulate and study the optimal control problem to drive the system from any initial state to a desired admissible terminal state in minimum time. We have established the existence of optimal solution using the Filippov's Existence Theorem and obtained the characteristics of the optimal control using the Pontryagin's Maximum Principle. Using the Hamiltonian minimization condition, we found the optimal control solution to be of bang-bang type with a possibility of admitting switches in the optimal trajectory in case the objective is to achieve biological conservation. To achieve pest eradication (prey elimination), we found the optimal control the continuous supply of to be of high quantity of additional food.
The ndings of this work are in line with some of the experimental observations and are also applicable in the ecological eld studies. For example, outcomes of studies in [48] state that increase in volume and concentration of arti cial diet sprays results in better pest eradication. This justi es the outcome of achieving prey elimination by providing a constant supply of high quality additional food in large quantity as stated in the results from section 5. Findings from [42] also state that pest elimination could be achieved by optimally providing high quality alternative diet to the generalist predators. However, in order to achieve biological conservation, the strategy of providing high quality of additional food in large quantities is not recommended because it could lead to apparent competition [26] and subsequently prey elimination. Consistent low quality additional food could also lead to predator elimination in some cases [30]. Thus, to achieve co-existence of species, an optimal strategy would be to switch the supply of additional food between the minimum and maximum quantity as stated in the results from section 4.
We have illustrated the theoretical ndings numerically by considering various cases. The type IV response is a generalization of the type II response where there is no inhibitory e ect in the latter response. One of the important ndings of this work is the role of inhibitory e ect in achieving bio-control. The numerical illustration ( gure 16) in section 7.3 shows that with increasing inhibitory e ect, more number of natural enemies are needed to control the pests and also the time taken to achieve the outcome increases. These ndings support the experimental observations from [22] which show that increasing inhibitory e ect could lead to reduced predation due to large defense mechanism. Thus, the strategy would be to increase natural enemies in order to attack the larger groups of prey species.
The outcomes of this work can be applied to biological conservation and bio-control of species. The illustrations establish the role of inhibitory e ect of prey in achieving the desired outcomes. The ndings also highlight the importance of quantity of additional food in the global dynamics of the system. These strategies can be implemented by eco-managers in the eld studies to achieve desired outcomes in minimum time. Proper vigilance and care must be taken while feeding the species because small errors could lead to undesirable outcomes.