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In this paper, a two species predator-prey model is developed where prey is affected by over-harvesting and drought and predator is affected by drought. The intention is to investigate the impact of over-harvesting and drought on predator-prey system, and suggest control strategies to alleviate the problem of loss of prey and predator species due to over-harvesting and drought. The control strategies suggested are creation of reserve areas with restriction of harvesting for controlling over-harvesting and construction of dams for mitigating drought effects. The results obtained from theoretical and numerical simulation of the predator-prey model with harvesting and drought without control strategies showed that, both harvesting and drought affect the predator-prey population negatively. However, the results obtained from numerical simulations of the model with control measures showed that, the use of control strategies one at a time encourages the increase of the prey and predator species to the optimal population size. Furthermore, the best result is obtained when control strategies, creation of reserve areas with restriction of harvesting and construction of dams are applied simultaneously .

Predator-prey system is a dynamical system that explains the dynamics of ecological systems in which two species prey and predator do interact. The interactions can be through predation, competition, parasitism, mutualism and detritivory [

The dynamic relationship between predators and their prey has been and continue to be one of the most important topic of discussion in ecology and mathematical ecology due to its universal existence and importance [

Harvesting in the predator-prey system is the killing of predator and prey species for the purpose of human consumption or trade. The problems of predator-prey systems with harvesting have been explored by many researchers, however most of them have put attention on the optimal harvesting guided entirely by economic profits from harvesting [

Drought can be defined as the absence of water for a long time to result in depletion of soil water and damage to plants [

Several studies have been conducted on optimal control strategies and management policies to keep and protect the ecological species such as those by [

Consider two different populations consisting of prey and predator species. The population densities are respectively represented by x and y. The dynamics of the species interactions is modelled using Holling type II functional response with the following assumptions.

1) In the absence of predators, harvesting and drought, prey population is assumed to grow logistically to the carrying capacity.

2) The rate of increase of the predators depend on the amount of prey biomass predators convert as food.

3) Predators depend on prey population as their only source of food.

From Holling type II functional response, the model is represented by the system of equations as below:

d x d t = r x ( 1 − x K 1 ) − α 1 x y x + A − α 2 x − d 1 x d y d t = − μ 1 y + β 1 x y x + A − d 2 y , x ( t ) > 0 , y ( t ) > 0. (2.1)

where x ( t ) represents the prey population density at time t, y ( t ) represents the predator population at time t, r > 0 stands for intrinsic growth rate of prey population, K 1 is the carrying capacity of the environment to prey species, α 1 > 0 represents the maximum per capita predation rate, β 1 > 0 is the conversion rate, α 2 > 0 is the rate of harvesting, d 1 > 0 is the death rate of prey species due to drought, μ 1 > 0 is the mortality rate of predator population, d 2 > 0 is the death rate of predator population due to drought and A > 0 represents the half saturation of a predator.

The solutions of the predator-prey model developed in (2.1) represent the populations of living individuals, and thus, they should be positive and bounded. The following lemma provides details:

Lemma: All the solutions of the system which start in IR + 2 are uniformly bounded.

Proof: We define a function

σ ( t ) = x ( t ) + α 1 y ( t ) β 1 (2.2)

where σ ( t ) represents the total population of the prey-predator species. Differentiating (2.2) with respect to time gives

d σ d t = d x d t + α 1 β 1 d y d t (2.3)

Substitute (2.1) into (2.3) and simplify yields

d σ d t = ( r − ( α 2 + d 1 ) ) x − r x 2 K 1 − α 1 β 1 ( μ 1 + d 2 ) y (2.4)

Choosing arbitrary constant say n 1 and applying it in (2.4) results into

d σ d t = ( r − ( α 2 + d 1 ) ) x − r x 2 K 1 − α 1 β 1 ( μ 1 + d 2 ) y + n 1 σ ( t ) − n 1 σ ( t ) d σ d t + n 1 σ ( t ) = ( r − ( α 2 + d 1 ) ) x − r x 2 K 1 − α 1 β 1 ( μ 1 + d 2 ) y + n 1 σ ( t ) = ( r − ( α 2 + d 1 ) ) x − r x 2 K 1 − α 1 β 1 ( μ 1 + d 2 ) y + n 1 ( x + α 1 β 1 y ) = ( r − ( α 2 + d 1 + n 1 ) ) x − r x 2 K 1 − α 1 β 1 ( μ 1 + d 2 − n 1 ) y ≤ ( r − ( α 2 + d 1 + n 1 ) ) x − r x 2 K 1

Thus,

d σ d t + n 1 σ ( t ) ≤ K 1 4 r ( r − ( α 2 + d 1 ) + n 1 ) 2 (2.5)

where

K 1 4 r ( r − ( α 2 + d 1 ) + n 1 ) 2 = m a x [ ( r − ( α 2 + d 1 + n 1 ) ) x − r x 2 K 1 ] .

By letting

K 1 4 r ( r − ( α 2 + d 1 ) + n 1 ) 2 = m 1

we get

d σ d t + n 1 σ ( t ) ≤ m 1 (2.6)

Solving Equation (2.6) by integrating factor I = e n 1 t yields

σ ( t ) e n 1 t ≤ m 1 n 1 + C e − n 1 t . (2.7)

At t = 0 , the Equation (2.7) becomes

σ ( t ) = m 1 n 1 + ( σ ( 0 ) − m 1 n 1 ) e − n 1 t . (2.8)

Thus,

0 ≤ σ ( t ) ≤ m 1 n 1 + ( σ ( 0 ) − m 1 n 1 ) e − n 1 t (2.9)

As t → ∞ Equation (2.9) becomes

0 ≤ σ ( t ) ≤ m 1 n 1 (2.10)

Therefore, σ ( t ) is bounded and from positivity of x and y,

0 ≤ x ≤ m 1 n 1 , 0 ≤ y ≤ m 1 n 1 . (2.11)

We start by setting the system of equations of model (2.1) equal to zero and by so doing we get the possible equilibrium points of the system as trivial equilibrium point ( A 0 ( 0,0 ) ) , predator free equilibrium point ( A 1 ( x * ,0 ) ) , prey free equilibrium point ( A 2 ( 0, y * ) ) and A 3 ( x * , y * ) the co-existence or interior equilibrium point. Thereafter we establish the conditions for existence of each equilibrium point of the system. The condition of existence of the equilibrium point A 0 ( 0,0 ) is trivial. The existence of the rest of the fixed equilibrium points is described as below:

1) Existence of A 1 ( x * ,0 ) with x * > 0 .

Let y = 0 , the system of equations of model (2.1) reduces to

x * ( r − r x * K 1 − α 2 − d 1 ) = 0 ,

which simplifies to

x * = K 1 ( r − α 2 − d 1 ) r .

Thus,

A 1 ( x * , 0 ) = A 1 ( K 1 ( r − α 2 − d 1 ) r , 0 ) . (2.12)

From the expression of x * we observe that harvesting and drought have negative impacts on the prey growth rate which eventually affects the prey population density. Moreover, for predator free equilibrium point A 1 ( x * ,0 ) to exist r − α 2 − d 1 > 0 which implies that r > α 2 + d 1 . Therefore, in the absence of predators the intrinsic growth rate of prey population should be greater than the sum of harvesting and drought rates. Increasing harvesting and drought rates, the prey population decreases which affects the survival of predator species. This is from the fact that predators depend on prey as their only source of food.

2) Existence of A 2 ( 0, y * ) with y * > 0 .

Let x * = 0 , the system of equations of model (2.1) reduces to y * ( − μ 1 − d 2 ) = 0 ,

from which we obtain y * = 0 . Thus,

A 2 ( 0 , y * ) = A 2 ( 0 , 0 ) . (2.13)

The result above concur with the assumption that predators depend on prey as their only source of food. Thus, in the absence of prey, predator population becomes extinct.

3) Co-existence equilibrium point A 3 ( x * , y * ) .

We equate the system of equations of model (2.1) to zero, and solving for x * and y * yields

x * = − A ( d 2 + μ 1 ) − β 1 + d 2 + μ 1 = A ( d 2 + μ 1 ) β 1 − d 2 − μ 1 , y * = A β 1 ( K 1 ( r − α 2 − d 1 ) ( β 1 − d 2 − μ 1 ) − A r ( d 2 + μ 1 ) ) K 1 α 1 ( β 1 − d 2 − μ 1 ) 2 . (2.14)

which gives the co-existence point (interior point).

From the values of interior point in (2.14), we see that predator mortality rate and drought affect the predator’s birth rate ( β 1 ) negatively which in turn results into negative effects on predator population density. On the other hand, the co-existence equilibrium point exist if β 1 − d 2 − μ 1 > 0 implying that β 1 > d 2 + μ 1 . Therefore, in the presence of both populations birth rate of predators should be greater than the sum of rates of mortality and drought. Furthermore, the intrinsic growth rate of prey should be greater than the sum of harvesting rate and drought rate of prey for co-existence equilibrium point to exist. However, increasing death rate of predator population results in the increase of prey population density.

The stability properties of the equilibrium points are analyzed by computing the Jacobian matrix and determining the eigenvalues of the Jacobian matrix of each fixed point. The equilibrium points are asymptotically stable if the real parts of the eigenvalues of each Jacobian matrix is negative. From the system of equations of model (2.1) the general Jacobian matrix of the equations is given by:

J ( A i ) = [ ∂ g 1 ∂ x ∂ g 1 ∂ y ∂ g 2 ∂ x ∂ g 2 ∂ y ]

which provides

J ( A i ) = [ r ( 1 − x * K 1 ) − r x * K 1 − α 1 y * x * + A + α 1 x * y * ( x * + A ) 2 − α 2 − d 1 − α 1 x * x * + A β 1 y * x * + A − β 1 x * y * ( x * + A ) 2 − μ 1 + β 1 x * x * + A − d 2 ] . (2.15)

1) For A 1 ( x * ,0 ) = A 1 [ K 1 ( r − α 2 − d 1 ) r ,0 ] ,

the Jacobian matrix evaluated at this equilibrium point is given by

J ( A 1 ) = [ − ( r − α 2 − d 1 ) − K 1 α 1 ( r − α 2 − d 1 ) A r + K 1 ( r − α 2 − d 1 ) 0 β 1 − A r β 1 A r + K 1 ( r − α 2 − d 1 ) − d 2 − μ 1 ] .

The eigenvalues of the Jacobian matrix J ( A 1 ) are − ( r − α 2 − d 1 ) and

β 1 − A r β 1 A r + K 1 ( r − α 2 − d 1 ) − d 2 − μ 1 .

Therefore, the equilibrium point A 1 ( x * ,0 ) is locally asymptotically stable if

it satisfies the following condition: β 1 − A r β 1 A r + K 1 ( r − α 2 − d 1 ) − d 2 − μ 1 < 0 .

which is negative if β 1 < d 2 + μ 1 . Thus, the predator free equilibrium point is locally asymptotically stable if predator population goes to extinction.

2) The corresponding Jacobian matrix of the equations evaluated at the equilibrium point A 2 ( 0 , y * ) = A 2 ( 0 , 0 ) is

J ( A 2 ) = [ r − α 2 − d 1 0 0 − μ 1 − d 2 ]

The eigenvalues for the Jacobian matrix J ( A 2 ) are r − α 2 − d 1 and − ( μ 1 + d 2 ) . If r > α 2 + d 1 , then the equilibrium point A 2 ( 0, y * ) is unstable saddle point. That is, to say the prey population is growing. However, if r < α 2 + d 1 then the prey population goes to extinction which results to extinction of predator population under the assumption that predators depends on prey as their only source of food. Thus, the prey free equilibrium point is locally asymptotically stable if prey population goes to extinction.

3) For co-existence fixed point A 3 ( x * , y * )

Consider the following Jacobian matrix

J ( A 3 ) = ( a 11 a 12 a 21 a 22 )

where

a 11 = ( d 2 + μ 1 ) ( ( − A r + K 1 ( r − α 2 ) ) β 1 + ( α 2 K 1 − ( A + K 1 ) r ) ( d 2 + μ 1 ) − M ) K 1 β 1 ( β 1 − d 2 − μ 1 ) a 12 = − α 1 ( d 2 + μ 1 ) β 1 a 21 = K 1 ( r − α 2 ) β 1 + ( α 2 K 1 − ( A + K 1 ) r ) ( d 2 + μ 1 ) − K 1 d 1 ( β 1 − d 2 − μ 1 ) α 1 K 1 a 22 = 0

and M = K 1 d 1 ( β 1 − d 2 − μ 1 )

The stability of this equilibrium point is stated using the trace/determinant technique as follows: Suppose the Jacobian matrix evaluated at the co-existence equilibrium point has the characteristic polynomial equation

λ 2 − P λ + Q = 0 , (2.16)

Such that P = t r ( J ( A 3 ) ) = a 11 + a 22 = a 11 and Q = D e t ( J ( A 3 ) ) = a 11 a 22 − a 21 a 12 = − a 21 a 12 .

Then, the co-existence equilibrium point is locally asymptotically stable or stable spiral if P < 0 and Q > 0 . However, the interior equilibrium point is a center (neutrally stable) if P = 0 and Q > 0 . Moreover, if the co-existence equilibrium point is locally asymptotically stable or stable spiral or neutrally stable, it implies that there is a stable dynamic relationship between predator and prey.

The global stability of equilibrium points A 1 and A 3 is shown by linearizing the system of equations of model (2.1) and defining appropriate Lyapunov function to separately describe each equilibrium point. The linearization process is done using Jacobian technique such that

d X i d t = J ( A i ) X i , (2.17)

where J ( A i ) is the Jacobian matrix and X i is a small perturbation on x i . Therefore the system of equations of model (2.1) reduces to the following linear system of equations;

d X d t = [ r ( 1 − x * K 1 ) − r x * K 1 − α 1 y * x * + A + α 1 x * y * ( x * + A ) 2 − α 2 − d 1 ] X − P d Y d t = [ β 1 y * x * + A − β 1 x * y * ( x * + A ) 2 ] X + [ − μ 1 + β 1 x * x * + A − d 2 ] Y (2.18)

where P = α 1 x * x * + A Y .

The Lyapunov function is chosen as

V ( X , Y ) = X 2 2 + Y 2 2 . (2.19)

The function V ( X , Y ) is a positive definite function since V ( X , Y ) > 0 for any values of ( X , Y ) and it is minimum at the origin. That is, V ( 0 , 0 ) = 0 . The time derivative of V ( X , Y ) is given by

d V ( X , Y ) d t = ∂ V ∂ X d X d t + ∂ V ∂ Y d Y d t . (2.20)

By substitution of (2.18) into (2.20) and differentiate V partially, we obtain the following

d V ( X , Y ) d t = X [ ( r ( 1 − x * K 1 ) − r x * K 1 − α 1 y * x * + A + α 1 x * y * ( x * + A ) 2 − α 2 − d 1 ) X − S ] + Y [ ( β 1 y * x * + A − β 1 x * y * ( x * + A ) 2 ) X + ( − μ 1 + β 1 x * x * + A − d 2 ) Y ] (2.21)

where S = α 1 x * x * + A Y .

1) For fixed point A 1 ( x * ,0 )

Here, Equation (2.12) is used in Equation (2.21) to get

d V ( X , Y ) d t = − ( r − α 2 − d 1 ) X 2 − K 1 α 1 ( r − α 2 − d 1 ) ( A + K ) r − K 1 α 2 − K d 1 X Y + ( β 1 − A r β 1 ( A + K 1 ) r − K 1 α 2 − K 1 d 1 − ( d 2 + μ 1 ) ) Y 2 (2.22)

From (2.22), the equilibrium point A 1 ( x * ,0 ) is globally asymptotically stable if the following condition is satisfied: That is,

( β 1 − A r β 1 ( A + K ) r − K 1 α 2 − K 1 d 1 − ( d 2 + μ 1 ) ) Y 2 < 0 ,

which is negative if β 1 < d 2 + μ 1 , implying that, the equilibrium point A 1 ( x * ,0 ) is globally stable if predators go to extinction.

2) For equilibrium point A 2 ( 0, y * ) .

In this subsection, we use Equation (2.13) in Equation (2.21) to obtain

d V ( X , Y ) d t = ( r − α 2 − d 1 ) X 2 − ( d 2 + μ 1 ) Y 2 (2.23)

From Equation (2.23), the fixed point A 2 ( 0, y * ) is globally asymptotically stable if ( r − α 2 − d 1 ) X 2 < 0 , which implies r < α 2 + d 1 . Thus, the prey free equilibrium point is globally asymptotically stable if prey population goes to extinction.

3) For steady state point A 3 ( x * , y * ) .

Here we substitute the values of interior point (2.14) into Equation (2.21) to obtain

d V ( X , Y ) d t = a 11 X 2 + ( a 12 + a 21 ) X Y , (2.24)

with usual notations for a 11 , a 12 and a 21 . Therefore the point is globally stable if the condition below holds,

d V ( X , Y ) d t = ( a 11 X 2 + ( a 12 + a 21 ) X Y ) < 0. (2.25)

We introduce into system of equations of model (2.1), time dependent control efforts on harvesting and drought to alleviate the loss of species in the predator-prey system. Let u 1 ( t ) represents over-harvesting control strategy (Creation of reserve areas with restriction of harvesting) and u 2 ( t ) represents the drought control strategy (construction of dams). Thus, the system of equations of model (2.1) becomes:

d x d t = r x ( 1 − x K 1 ) − α 1 x y x + A − ( 1 − u 1 ( t ) ) α 2 x − ( 1 − u 2 ( t ) ) d 1 x d y d t = − μ 1 y + β 1 x y x + A − ( 1 − u 2 ( t ) ) d 2 y (3.1)

Here we construct an objective function that provides the optimal population size of the prey-predator species at minimum costs for over-harvesting and drought control strategies. The objective function takes the following form, that is

J ( u i ) = max [ G ( x i ( T 1 ) , T 1 ) − ∫ T 0 T 1 ( F ( x i ( t ) , u i ( t ) , t ) ) d t ] (3.2)

subject to

d x i d t = g i ( t , u i ( t ) , x i ( t ) ) ,

where x i ( T 0 ) = x i 0 and 0 ≤ u i ≤ 1 for t ∈ [ T 0 , T 1 ] . The terms G ( x i ( T 1 ) ) and F ( x i ( t ) , u i ( t ) , t ) represent the predator-prey populations to be optimized at terminal time of control and total cost of control respectively.

Therefore, from (3.2) the objective function becomes:

J ( U ) = max u 1 , u 2 [ D 1 x ( T 1 ) + D 2 y ( T 1 ) − ∫ 0 T 1 ( C 1 u 1 2 2 + C 2 u 2 2 2 ) d t ] (3.3)

subject to

d x d t = r x ( 1 − x K 1 ) − α 1 x y x + A − ( 1 − u 1 ( t ) ) α 2 x − ( 1 − u 2 ( t ) ) d 1 x d y d t = − μ 1 y + β 1 x y x + A − ( 1 − u 2 ( t ) ) d 2 y

x ( T 0 ) = x 0 , y ( T 0 ) = y 0 and 0 ≤ u 1 ≤ 1,0 ≤ u 2 ≤ 1 for t ∈ [ 0, T 1 ] ; u 1 , u 2 ∈ U .

However, the terms D 1 x ( T 1 ) and D 2 y ( T 1 ) represent the prey and predator

populations to be optimized at terminal time of control and C 1 u 1 2 2 and C 2 u 2 2 2

are total control costs for over-harvesting and drought respectively. The costs weights C 1 , C 2 and state weights D 1 , D 2 are all positive constants. In here, the aim is to pick up u 1 * and u 2 * such that,

J ( u 1 * , u 2 * ) = m a x U ( J ( u 1 , u 2 ) ) , (3.4)

with 0 ≤ u 1 ≤ 1,0 ≤ u 2 ≤ 1 for t ∈ [ 0, T 1 ] .

The aim is to show that, the optimal control problem formulated in (3.3) has at least one solution before trying to solve for the optimal control values. The following theorem explains. Theorem: Given an optimal control problem in (3.3) with u 1 and u 2 as control variables, then there exist u 1 * , u 2 * ∈ U (optimal control set) such that,

J ( u 1 * , u 2 * ) = m a x U ( J ( u 1 , u 2 ) ) , (3.5)

Proof: The proof for the existence of optimal control provided by [

1) Model Equation (3.1) with control are linear in control variables.

2) The control U is convex, closed and bounded set.

3) The integrand − ( C 1 u 1 2 2 + C 2 u 2 2 2 ) of the objective function (3.3) is concave

in U .

The calculation of the optimal control strategy is done by the application of Pontryagin’s maximum principle (PMP) which provides the Hamiltonian H and necessary conditions with which the optimal control and the co-state variables must satisfy. From the objective function (3.3), the Hamiltonian H is given by

H = − ( C 1 u 1 2 2 + C 2 u 2 2 2 ) + R 1 [ r x ( 1 − x K 1 ) − α 1 x y x + A − ( 1 − u 1 ( t ) ) α 2 x − ( 1 − u 2 ( t ) ) d 1 x ] + R 2 [ − μ 1 y + β 1 x y x + A − ( 1 − u 2 ( t ) ) d 2 y ] , (3.6)

where R 1 and R 2 represents the co-state variables or adjoint variables. Using Pontryagin’s maximum principle (PMP) [

Proposition: Let u 1 * and u 2 * be optimal control variables that optimize the objective function in (3.3), then there exist adjoint variables R 1 and R 2

satisfying the adjoint condition d R i d t = ∂ H ∂ x i with R i ( T ) = D i ; i = 1 , 2 .

Proof: Using the adjoint condition, the set of adjoint equations becomes:

d R 1 d t = − ∂ H ∂ x = − R 1 [ r ( 1 − x K 1 ) − r x K 1 − α 1 y x + A + α 1 x y ( x + A ) 2 − ( 1 − u 1 ( t ) ) α 2 − ( 1 − u 2 ( t ) ) d 1 ] − R 2 [ β 1 y x + A − β 1 x y ( x + A ) 2 ] d R 2 d t = − ∂ H ∂ y = − R 1 [ − α 1 x x + A ] − R 2 [ − μ 1 + β 1 x x + A − ( 1 − u 2 ( t ) ) d 2 ] (3.7)

with transversality conditions:

R 1 ( T 1 ) = d d x ( D 1 x ( T 1 ) ) = D 1 , R 2 ( T 1 ) = d d y ( D 2 y ( T 1 ) ) = D 2 . (3.8)

Using optimality condition,

∂ H ∂ u i = 0 at u i * , we have ∂ H ∂ u 1 at u 1 * and ∂ H ∂ u 2 at u 2 * . Thus, ∂ H ∂ u 1 = − C 1 u 1 * + R 1 α 2 x = 0 which gives

u 1 * = R 1 α 1 x C 1 (3.9)

Also,

∂ H ∂ u 2 = − C 2 u 2 * + R 1 d 1 x + R 2 d 2 y = 0

Thus,

u 2 * = R 1 d 1 x + R 2 d 2 y C 2 (3.10)

Solving for x, y, R 1 and R 2 , we substitute u 1 * and u 2 * in the following system of equations below:

d x d t = r x ( 1 − x K 1 ) − α 1 x y x + A − ( 1 − u 1 ( t ) ) α 2 x − ( 1 − u 2 ( t ) ) d 1 x , x ( 0 ) = x 0 d y d t = − μ 1 y + β 1 x y x + A − ( 1 − u 2 ( t ) ) d 2 y , y ( 0 ) = y 0 d R 1 d t = − R 1 [ r ( 1 − x K 1 ) − r x K 1 − α 1 y x + A + α 1 x y ( x + A ) 2 − ( 1 − u 1 ( t ) ) α 2 − W 1 ]

− R 2 [ β 1 y x + A − β 1 x y ( x + A ) 2 ] , R 1 ( T 1 ) = D 1 d R 2 d t = − R 1 [ − α 1 x x + A ] − R 2 [ − μ 1 + β 1 x x + A − ( 1 − u 2 ( t ) ) d 2 ] , R 2 ( T 2 ) = D 2 , (3.11)

where, W 1 = ( 1 − u 2 ( t ) ) d 1 .

Taking back the values obtained in (3.11) into the expressions of u 1 * and u 2 * in (3.9) and (3.10) we get the characterization of the optimal control as:

u 1 * = min { 1 , max ( 0 , R 1 α 2 x C 1 ) } u 2 * = min { 1 , max ( 0 , R 1 d 1 x + R 2 d 2 y C 2 ) } , (3.12)

where R 1 and R 2 are the solutions of the system of adjoint Equation (3.7).

We verify the equilibrium points of system equations of model (2.1), using phase diagrams representation. The aim is to show the dynamical behavior of the equilibrium points discussed in the theoretical part. Using parameter values as in

time, on prey population density. The result shows that, there is a rapid decrease in prey population. The rapid decrease in prey population is due to the combinations of the problems (harvesting and drought) on prey species both with strong negative impact on the prey species.

predator population density while the effects are on prey individuals is due to dependence of predators on prey as their only source of food. Thus, decreasing prey population rapidly results into rapid decrease in predator population density.

From

that drought affect the population negatively. From

The optimal control strategy can numerically be solved by several numerical techniques using parameter values. A forward-backward sweep method (FBSM) is one of the numerical techniques that can be used to solve an optimal control problem. The technique uses Runge-Kutta 4th order numerical method (RK4) in solving the optimal control problem. The following scholars [

1) Make an initial guess for control vector and use initial conditions for state vector to solve for the state variables forward in time using Runge-Kutta 4th order numerical method (RK4).

2) Using the new sate values, transversality conditions and the guessed values for control vector, solve for adjoint vector backward in time using RK4.

3) The obtained values for state and adjoint variables are entered on the characterization of the optimal control (3.12) to update the control vector which becomes a new value for the control vector.

4) Stop the process when the values of the control variables in the current and previous iterations are sufficiently close.

The investigation of the impacts of the control efforts on the predator-prey system is studied numerically through different combinations of the control strategies. The following

Control strategy | Description |
---|---|

Strategy I | Creation of reserve areas with restriction of harvesting for controlling over-harvesting |

Strategy II | Construction of dams for mitigating drought effects |

Strategy III | Combination of control strategy I and II |

Parameter name | Symbol | Value | Source |
---|---|---|---|

Predation rate | α 1 | 0.367 | Estimated |

Conversion rate | β 1 | 0.25 | [ |

Intrinsic growth rate | r | 1 | [ |

Predator’s half saturation | A | 1 | [ |

Predator’s mortality rate | μ 1 | 0.01 | [ |

Carrying capacity | K 1 | 200 | Estimated |

Harvesting rate | α 2 | 0.5 | [ |

Prey’s drought rate | d 1 | 0.15 | [ |

Predator’s drought rate | d 2 | 0.139 | Estimated |

According to [

From

u_{1} is shown by using it to optimize the objective function J ( U ) while setting the control strategy for drought u_{2} equal to zero. The result shows that, at the final time of control the prey population increases from 24 to about 128 individuals. These results indicate that, over-harvesting has strong negative impact on the population dynamics of prey-predator population densities.

In _{1} is used to optimize the objective function J ( U ) while optimal control strategy for drought, u_{2} is set to zero. The result shows that, Despite of predators having enough food due to increased prey species as shown in

From _{2} for mitigating drought effects is shown by using the control strategy to optimize the objective function J ( U ) while setting the control strategy for over-harvesting u_{1} equal to zero. The result shows that after control there is increase in small number in prey population from about 24 (final state before control) to about 30 species only, at the final time of control. The increase in small number of prey population density while drought has been controlled is due to increased predators (

In _{2} to predator-prey population densities. We observe that, the predator population density has increased from 24 individuals to about 76 individuals. The strategy is much successful in optimizing predator population. This is due to the assumption that, drought is the only external factor that affects predator population density.

From _{1} and u_{2} are shown by using them simultaneously to optimize the objective function J ( U ) . The result of control showed that, prey population increases from 24 to about 150 species. Thus, applying both control strategies at a time to predator-prey densities, encourages the growth of prey in the predator-prey system.

From

The aim of this study was to investigate the effects of over-harvesting and drought, and to assess the impact of control measures to these external forces. We developed a predator-prey model which incorporates harvesting and drought without control. Thereafter we carried out some analysis (theoretically and numerically) of the developed model to describe how harvesting and drought affect the prey and predator species. Furthermore, the model was modified to include the time dependent control efforts on over-harvesting and on drought. The numerical analysis to investigate the impact of the time dependent control

efforts on the predator-prey model was also carried out.

However, from the numerical simulations we observe that, using control strategies one at a time or two at a time encourages the increase of prey and predator population densities to the optimal population size. Furthermore, the best results is attained when the control strategies are applied at a time. Thus, in order to keep and protect the prey and predator species from over-harvesting and drought, the stakeholders should choose to apply for control strategies to both over-harvesting and drought simultaneously.

The authors declare no conflicts of interest regarding the publication of this paper.

Mapunda, A., Mureithi, E., Shaban, N. and Sagamiko, T. (2018) Effects of Over-Harvesting and Drought on a Predator-Prey System with Optimal Control. Open Journal of Ecology, 8, 459-482. https://doi.org/10.4236/oje.2018.88028