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Hindawi International Journal of Mathematics and Mathematical Sciences Volume 2019, Article ID 5079171, 9 pages https://doi.org/10.1155/2019/5079171 Research Article Dynamics of Leslie-Gower Pest-Predator Model with Disease in Pest Including Pest-Harvesting and Optimal Implementation of Pesticide Agus Suryanto and Isnani Darti Department of Mathematics, University of Brawijaya, Jl. Veteran, Malang , Indonesia Correspondence should be addressed to Agus Suryanto; suryanto@ub.ac.id Received 28 January 2019; Accepted 30 May 2019; Published 18 June 2019 Academic Editor: Irena Lasiecka Copyright © 2019 Agus Suryanto and Isnani Darti. iTh s is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We propose a model which describes the interaction between pest and its natural predator. We assume that pest can be infected with diseases or pathogens such as bacteria, fungi, and viruses. eTh model is constructed by combining the Leslie-Gower model and S-I epidemic model. It is also considered the eeff cts of pest harvesting. Harvesting in this case is intended to take a number of pests as one of the pest population control strategies. The proposed model will be analyzed dynamically to study its qualitative behaviour. The dynamical analysis includes the determination of all possible equilibrium points and their stability properties. Furthermore we also discuss the implementation of pesticide control where its optimal strategy is determined by Pontryagin’s maximum principle. To support our analytical studies, we perform some numerical simulations and their interpretation. 1. Introduction in prey on the dynamics of prey and predator. To implement an integrated pest control we need to know what kind of natural enemy should be released, how many pest should be Pest is organism that can be very damaging to agricultural production, which is a serious problem throughout the world killed or trapped, and how much biopesticides (pathogens) [1]. Therefore, many researchers are interested in developing or chemical pesticides should be sprayed. To regulate the effective pest control methods. The classical method for pest control strategy, it is very useful to study the population eradicating pests is to spray chemical pesticides, which can changes of both pest and predator in a long time. quickly kill a significantly number of pests. However, the The interaction of pest with its natural predator with use of chemical pesticides is known to have side effects disease in pest is theoretically described by ecoepidemic such as pests becomes resistant to pesticides or accumulation model, which is a combination of the ecological model and the epidemiological model. The ecoepidemic model has been of chemical toxic residues in human food chain [2, 3]. To overcome the side eeff cts of the use of chemical pesticides, widely proposed and investigated [8–13]. Anderson and May pest control is carried out biologically which is expected [8] have investigated the interaction of pest (susceptible and infected) with its predator and observed that the infectious to maintain the ecological environment and provide pest control solutions in a sustainable manner. An example of disease within animal and plant communities may destabilize biological control is by releasing natural enemies to kill pest the system. Recently, Ghosh et al. [10], Jana and Kar [11], and or maintain pest density to remain below the threshold of eco- Kar et al. [12] proposed mathematical models which describe nomic or ecological damage. An alternative biological control the interaction of pest and its predator where pest population is by providing pathogens (bacteria, fungi, and viruses) to is controlled by releasing infectious disease, natural enemy, infect or to kill the pest [4–6]. Recently Meng et al. [7] also and the use of chemical methods (pesticides). Here, the use of studied the combination effects of harvesting and pathogens pesticide is optimized by Pontryagin’s maximum principle to 2 International Journal of Mathematics and Mathematical Sciences minimize the implementation cost for the pesticide as well as Table 1: Biological interpretation of parameters. its side effects, while effectively reducing the pest population. Parameter Biological interpretation The models proposed in [10–12] are based on Lotka-Volterra 𝑟 Intrinsic growth rate of 𝑆 model where the growth rate of predator is proportional to 𝐾 Carrying capacity of 𝑆 the rate of predation. If the predator is assumed to grow logistically where its carrying capacity depends on the size of 𝛽 Infection rate prey, then we can apply a Leslie-Gower model. Ecoepidemic 𝑐 Death rate of 𝐼 induced by the disease models based on the Leslie-Gower equation have been pro- 𝑐 Predation rate posed by a number of researchers [14–19]. Zhou et al. [19] and 𝑐 Competition rate of 𝑌 Sharma and Samanta [16] considered an ecoepidemic model 𝑎 Intrinsic growth rate of 𝑌 based on Leslie-Gower model with infectious disease in prey 𝑘 Constant of environment protection to 𝑆 where the functional response follows the Holling type II. 𝑘 Constant of environment protection to 𝑌 They considered that predator only consumes infected preys 𝛼 Harvesting rate of 𝑆 and the transmission of disease follows a bilinear incidence rate. The prey harvesting effect on model proposed in [19] has been investigated by Purnomo et al. [15]. Recently, Suryanto Proof. Based on system (1a), (1b), and (1c), we directly get [17] has also modified model in [19] by assuming the saturated incidence rate. Moreover, Suryanto et al. [18] also studied the 𝑆 (𝑧 )+𝐼 (𝑧 ) 𝑆 (𝑡 )=𝑆 exp ∫ [𝑟 (1 − )−𝛽𝐼 (𝑧 ) Leslie-Gower ecoepidemic model where predator only eats susceptible prey. In this paper, we develop model in [18] by including the effects of pest harvesting. The pest harvesting is 𝑐 𝑌 (𝑧 ) intended to take a number of pests. Hence, the pest harvesting − −𝛼]𝑧𝑑 > 0, 𝑘 +𝑆 (𝑧 ) is considered as one of the pest population control strategies. (3) The ecoepidemic model in this article is then given by 𝐼 (𝑡 )=𝐼 exp ∫ [𝛽𝑆 (𝑧 )−𝑐]𝑧𝑑 > 0, 𝑆+ 𝐼 𝑐 𝑆𝑌 1 0 =𝑟(1− )𝑆 − 𝐼𝑆𝛽 − −𝛼𝑆 (1a) 𝐾 𝑘 +𝑆 𝑐 𝑌 (𝑧 ) 𝑌 (𝑡 )=𝑌 exp ∫ [𝑎 − ]𝑑𝑧 > 0. 0 2 (1b) =𝛽𝐼𝑆−𝑐𝐼 𝑘 +𝑆 (𝑧 ) It is clear that every solution of initial value problem (1a), 𝑐 𝑌 (1b), and (1c)-(2) will always be positive =(𝑎 − )𝑌 (1c) 𝑘 +𝑆 Lemma 2. (i) e number of preys is bounded above. with initial conditions (ii) e number of predators is bounded above. (iii) System (a), (b), and (c) is uniformly bounded. 𝑆 0 =𝑆 >0, () Proof. (i) For 𝑆(0) = 0, we directly get the result. For(𝑆 0)>0, 𝐼 (0)=𝐼 >0, (2) the positivity solution gives 𝑆(𝑡) > 0. Furthermore, from (1a) 𝑌 (0)=𝑌 >0. and (1b) we get Here, 𝑆 and 𝐼 are the susceptible and infected pest, respec- ≤𝑟(1− )𝑆, tively, while 𝑌 represents the predator. All parameters in (1a), (1b), and (1c) are assumed to be positive and their biological (4) interpretation are given in Table 1. Notice that model (1a), 𝑑 (𝑆+ 𝐼 ) 𝑆+ 𝐼 ≤𝑟(1− )𝑆, (1b), and (1c) is a combination of Leslie-Gower model and𝑆−𝐼 epidemic model with bilinear incidence rate. Furthermore, which imply that lim sup𝑆≤ 𝐾 and lim sup (𝑆+ →∞ →∞ the susceptible pest is continuously harvested with rate 𝛼 . 𝐼) ≤ 𝐾 . (ii) The result is directly obtained for 𝑌(0) = 0. Similarly, 2. Dynamical Properties for 𝑌(0) > 0we have 𝑌(𝑡) > 0 . From (1c) and previous result, we obtain .. Positivity and Boundedness. It is obvious that the popula- tion is impossibleto havenegative value,while thebounded- 𝑐 𝑌 ≤(𝑎 − )𝑌. (5) ness of solutions can be understood as a natural limitation for 𝑘 +𝐾+𝜀 growth as a consequence of limited resources. Positivity and Therefore, we have that lim sup𝑌≤ (𝑘 +𝐾+𝜀)𝑎 /𝑐 . boundedness of solutions of system (1a), (1b), and (1c) will be →∞ 2 2 2 Since𝜀>0 is arbitrary, we can conclude that lim sup𝑌≤ presented as follows. →∞ (𝑘 +𝐾)𝑎 /𝑐 . 2 2 2 Lemma 1. Every solution of system (a), (b), and (c) with (iii) We set 𝑝(𝑡) = ( 𝑆 𝑡) + 𝐼(𝑡) + ( 𝑌 𝑡) and calculate the initial condition () will be positive ∀𝑡 > 0 . derivative of 𝑝(𝑡) with respect to 𝑡 to get 푡㨀 푡㨀 𝑑𝑡 𝑑𝑌 푡㨀 푡㨀 𝑑𝑡 𝑑𝑡 𝑑𝑆 𝑑𝑡 𝑑𝑌 𝑑𝑡 𝑑𝐼 𝑑𝑡 𝑑𝑆 International Journal of Mathematics and Mathematical Sciences 3 () 𝑆+ 𝐼 𝑐 𝑆𝑌 1 .. Equilibria of the System. System (1a), (1b), and (1c) has six =𝑟(1 − )𝑆 − −𝛼𝑆 − 𝑐𝐼 𝐾 𝑘 +𝑆 biologically feasible equilibria: (6) 𝑐 𝑌 (i) 𝐸 = (0,0, 0) , where all populations are extinct. The +(𝑎 − )𝑌. 𝑘 +𝑆 extinction of all populations equilibrium (𝐸 )always exists. Now, for any 𝑞<𝑐 we have that (ii) 𝐸 = (𝐾(𝑟 − )𝛼 /,0 𝑟 ,0) , where only susceptible prey survives. The survival of susceptible prey equilibrium (𝐸 )isfeasible if 𝑟> 𝛼 , i.e., when the intrinsic () +𝑞𝑝 𝑡 ≤(𝑟 +𝑞)𝑆+ (𝑎 +𝑞).𝑌 (7) growth rate of prey is larger than the harvesting rate () of susceptible prey. (iii) 𝐸 = (0,0, 𝑎 𝑘 /𝑐 ), where prey is extinct and predator 3 2 2 2 For a given 𝜀>0 ,wehave 𝑡 >0 such that survives. The prey-free equilibrium ( 𝐸 )is always feasible. (𝑡 ) ∗ ∗ (iv) 𝐸 =(𝑆 ,𝐼 ,0), where the predator is extinct with (8) +𝑞𝑝 (𝑡 )≤𝜎+,𝜀 4 4 4 ∗ ∗ 𝑆 =𝑐/𝛽 and 𝐼 = (𝐾(𝑟 𝛽 − ) 𝛼 − 𝑐𝑟)/(𝛽(𝛽𝐾 + 𝑟)) . 4 4 The predator-free equilibrium ( 𝐸 )isfeasibleif(−𝑟 for any 𝑡> 𝑡 and 𝜎 = (𝑟 + 𝑞)𝐾 + (𝑎 + 𝑞)((𝑘 +𝐾)𝑎 /𝑐 ). 𝛼) > 𝑐𝑟 . 0 2 2 2 2 Applying Gron ¨ wall’s differential inequality, we get ∗ ∗ (v) 𝐸 =(𝑆 ,0,𝑌 ), where there is not infected prey. Here 5 5 5 𝑆 =(−(𝑎𝑐 𝐾+𝑐 𝐾(𝛼 − 𝑟) + 𝑐 𝑘 𝑟) + 𝐷)/2𝑐 𝑟, 𝐷= 2 1 2 2 1 2 𝑝 (𝑡 )≤𝑝(𝑡 )exp (−𝑞(𝑡 − 𝑡 )) (𝑎 𝑐 𝐾+𝑐 𝐾(𝛼−𝑟)+𝑐 𝑘 𝑟) −4𝑐 (𝑎 𝑘 𝑐 +𝑐 𝑘 (𝛼−𝑟)), 0 0 2 1 2 2 1 2 2 2 1 2 1 ∗ ∗ and 𝑌 =𝑎 (𝑆 +𝑘 )/𝑐 .Thefreeof infected prey (9) 2 2 2 𝜎+ 𝜀 5 5 + (1 −exp (−𝑞 (𝑡 − 𝑡 ))) , 0 equilibrium (𝐸 )existsif 𝑟− 𝛼 > 𝑐 𝑎 𝑘 /(𝑐 𝑘 ). 5 1 2 2 2 1 ∗ ∗ ∗ ∗ (vi) 𝐸 =(𝑆 ,𝐼 ,𝑌 ), where prey and predator coexist. ∗ ∗ ∗ Here,𝑆 =𝑐/,𝛽 𝑌 =𝑎 (𝛽𝑘 +𝑐)/(𝛽𝑐 )and 𝐼 = ((𝑘+ 2 2 2 1 and for 𝑡→ ∞ and taking 𝜀→0 ,weget ∗ ∗ 𝑆 )(𝐾(𝛽 𝑟−𝛼)−𝑐)−𝑐 𝑟 𝑌 )/(𝑐+𝑘𝛽 )(𝛽𝐾+𝑟) .The 1 1 coexistence equilibrium is feasible if 𝐼 >0. .. Stability of Equilibria. The local stability of equilibrium lim sup 𝑝 (𝑡 )≤ . (10) →∞ point of system (1a), (1b), and (1c) is determined by the eigenvalues of Jacobian matrix of system (1a), (1b), and (1c). ̂ ̂ ̂ ̂ Hence, system (1a), (1b), and (1c) is uniformly bounded. Here, the Jacobian matrix at an equilibrium point𝐸= ( 𝑆, 𝐼, 𝑌) is ̂ ̂ ̂ ̂ 2𝑆+ 𝐼 𝑐 𝑘 𝑌 𝑟 𝑐 𝑆 1 1 1 ̂ ̂ 𝑟(1 − )− 𝛽 𝐼− −𝛼 −( +𝛽) 𝑆− 2 2 𝐾 𝐾 ̂ ̂ (𝑆+ 𝑘 ) (𝑆+ 𝑘 ) 1 1 ( ) ̂ ̂ 𝛽𝐼𝛽 𝑆− 𝑐 0 𝐽( 𝐸) = ( ) . (11) ̂ ̂ 𝑐 𝑌 2𝑐 𝑌 2 2 0− +𝑎 ̂ (𝑆+ 𝑘 ) (𝑆+ 𝑘 ) ( 2 ) 𝑐 𝑎 𝑘 The direct evaluation of the Jacobian matrix at 𝐸 ,𝐸 ,𝐸 ,and 1 2 2 1 2 3 𝑟− 𝛼− 00 𝐸 is, respectively, 𝑐 𝑘 2 1 0−𝑐 0 𝐽(𝐸 )=( ) 𝑟− 𝛼 0 0 0−𝑎 0−𝑐 0 𝐽(𝐸 )= ( ), 00 𝑎 (12) 𝐽(𝐸 ) 2 and 𝐽(𝐸 ) (𝛽𝐾 + 𝑟) (𝑟− 𝛼 ) 𝑐 𝐾 (𝛼− 𝑟 ) 𝛼− 𝑟 − 2𝑐 + 𝐼𝛽 𝑐(𝛽 K +𝑟) 𝑐𝑐 𝑟 𝑘 𝑟+𝐾 (𝑟−𝛼 ) 4 ∗ 1 𝑟(1 − )− 𝛼−𝐼𝛽 − − (13) =( (𝑟−𝛼 )−𝑐𝑟 ), +𝑐 0 0 =( ). 00 𝑎 00 𝑎 2 2 𝛽𝐼 𝛽𝑘 𝛽𝐾 𝛽𝐾 𝛽𝐾 푡㨀 𝛽𝐾 𝑟𝐾 𝛽𝐾 𝑑𝑡 𝑑𝑝 𝑑𝑡 𝑑𝑝 𝑑𝑡 𝑑𝑝 4 International Journal of Mathematics and Mathematical Sciences All eigenvalues of 𝐽(𝐸 ), 𝐽(𝐸 ),𝐽(𝐸 ),and 𝐽(𝐸 )are very easy the pest-free equilibrium is stable. The sufficient condition for 1 2 3 4 to be determined and analyzed. An equilibrium is asymptot- the pest-free equilibrium to be globally asymptotically stable ically (locally) stable if the real parts of all eigenvalues of its is given by the following theorem. Jacobian matrix are negative, and consequently we have the Theorem 6. If 𝑟< 𝛼 + 𝑐 𝑎 𝑘 /𝑐 (𝑘 +𝐾) ,then the pest-free following result. 1 2 2 2 1 equilibrium (𝐸 ) is globally asymptotically stable. Theorem 3. Equilibrium points 𝐸 ,𝐸 ,and 𝐸 are always 1 2 4 Proof. From Theorem 3, we know that if 𝑟<𝛼+𝑐 𝑎 𝑘 /𝑐 (𝑘 + unstable. If 𝑟<𝛼 + 𝑐 𝑎 𝑘 /(𝑐 𝑘 ),then 𝐸 is locally 1 2 2 2 1 1 2 2 2 1 3 𝐾) < 𝛼 + 𝑐 𝑎 𝑘 /𝑐 𝑘 then the equilibrium 𝐸 is locally asymptotically stable. 1 2 2 2 1 3 asymptotically stable. To show the global stability of 𝐸 ,we The evaluation of Jacobian matrix (11) at 𝐸 gives have to prove that lim 𝑆(𝑡) = 0, lim 𝐼(𝑡) = 0,and 5 →∞ →∞ lim 𝑌(𝑡) = 𝑎 𝑘 /𝑐 . From (1c), we notice that →∞ 2 2 2 𝐽(𝐸 ) 𝑐 𝑌 𝑐 𝑌 2 2 =(𝑎 − )𝑌 ≥ (𝑎 − )𝑌, (17) ∗ ∗ ∗ ∗ 2 2 2𝑆 𝑐 𝑘 𝑌 𝑐 𝑆 𝑘 +𝑆 𝑘 1 1 1 5 5 5 ∗ 5 2 2 𝑟(1 − )− 𝛼− − −𝛽𝑆 − 2 5 ∗ ∗ (14) 𝐾 𝐾 𝑘 +𝑆 (𝑘 +𝑆 ) 1 1 5 =( ∗ ). which gives that lim inf𝑌≥𝑎 𝑘 /𝑐 .Based on this 0𝛽𝑆 −𝑐 0 →∞ 2 2 2 result, Lemmas 1 and 2(i), and (1a), we have that 𝑑𝑆/𝑑𝑡 ≤ 𝑎 /𝑐 0−𝑎 2 2 2 − 𝑐 𝑆𝑌/(𝑘 +𝑆)−𝛼𝑆 ≤ (𝑟−𝑐 𝑎 𝑘 /𝑐 (𝑘 +𝐾)−𝛼)𝑆 . 1 1 1 2 2 2 1 Then, lim 𝑆(𝑡) = 0 if 𝑟< 𝛼+ 𝑐 𝑎 𝑘 /𝑐 (𝑘 +𝐾) .For any The characteristics equation of 𝐽(𝐸 )is given by the following →∞ 1 2 2 2 1 𝜀> 0 , there exists some sufficiently large 𝑡 >0 such that cubic equation: −𝜀 < (𝑡 𝑆 ) < 𝜀 for 𝑡≥ 𝑡 .Using (1b), we obtain 3 2 𝜆 +𝐴 𝜆 +𝐴 𝜆+ 𝐴 =0, (15) 1 2 3 (18) (−𝛽𝜀 − 𝑐) 𝐼 ≤ ≤(𝜀𝛽− 𝑐)𝐼. ∗ ∗ 2 ∗ ∗ where 𝐴 =−(𝑐 𝑌 /(𝑘 +𝑆 )−𝑟/𝐾)𝑆 −(𝑆𝛽 −𝑐)+𝑎 ,𝐴 = 1 1 5 1 5 5 5 2 2 ∗ ∗ ∗ 2 ∗ ∗ ∗ ∗ 2 𝑆 (𝑐 𝑌 /(𝑘 +𝑆 ) − /𝑟 𝐾)(𝛽𝑆 −𝑐)−𝑎 𝑆 (𝑐 𝑌 /(𝑘 +𝑆 ) − 5 1 5 1 5 5 2 5 1 5 1 5 By taking 𝜀→0 , we get the following comparison equation: ∗ 2 ∗ ∗ 𝑟/𝐾) + 𝑐 𝑆 𝑎 /𝑐 (𝑘 +𝑆 )−𝑎 (𝛽𝑆 −𝑐) and 1 2 1 2 5 2 5 5 ∗ ∗ ∗ ∗ 2 𝐴 =𝑎 𝑆 (𝛽𝑆 −𝑐)((𝑐𝑌 /(𝑘 +𝑆 )−𝑟/𝐾)−𝑐 𝑎 /𝑐 (𝑘 + () 3 2 1 1 1 2 2 1 5 5 5 5 (19) =−𝑐𝑀 (𝑡 ). ∗ ∗ ∗ 2 ∗ 𝑆 )). It is obvious that if𝑐 𝑌 /(𝑘 +𝑆 ) <𝑟/𝐾 and <𝑐 , 5 1 5 1 5 5 then 𝐴 >0,𝐴 >0, 𝐴 >0 and also 1 2 3 Obviously 𝑀= 0 is the only equilibrium of (19). Further- ∗ ∗ more, 𝑀= 0 is globally asymptotically stable equilibrium. 𝐴 =𝐴 𝐴 −𝐴 =𝑆 (𝛽𝑆 −𝑐) 4 1 2 3 5 5 Based on the comparison theorem, we get lim 𝐼(𝑡) = 0. →∞ Using similar argument, we can also show that lim 𝑌(𝑡) = 𝑐 𝑌 𝑟 →∞ 1 5 ∗ ⋅(−( − )−(𝑆𝛽 −𝑐)+𝑎 ) 5 2 𝑎 𝑘 /𝑐 . 2 2 2 (𝑘 +𝑆 ) 1 5 .. Numerical Simulations and Discussion. To illustrate the 𝑐 𝑌 dynamical behaviour of system (1a), (1b), and (1c), we per- ⋅( − ) (16) (𝑘 +𝑆 ) form some numerical simulations using hypothetical value 1 5 of parameters. First, we set parameters 𝑟 = 1,𝐾 = 1, 𝛽 = ∗ 2 ∗ 𝑐 𝑆 𝑎 𝑐 𝑌 1 1 0.1, 𝑐 = 0.01,𝑐 =0.5,𝑐 = 0.25, 𝑘 =1,and 𝑘 =0.5. 5 2 5 1 2 1 2 − (( − )− 𝑎 ) ∗ 2 𝑐 (𝑘 +𝑆 ) 𝐾 Using these parameters, we show in Figure 1 the stability (𝑘 +𝑆 ) 2 1 5 1 5 areas of 𝐸 ,𝐸 ,and 𝐸 with respect to 𝑎 and 𝛼 .Itis shown 3 5 2 ∗ 2 2 ∗ that, for a relatively small intrinsic predator growth rate (𝑎 ) +𝑎 (𝛽𝑆 −𝑐) −𝑎 (𝛽𝑆 −𝑐) > 0. 2 5 2 5 and a relatively low harvesting rate of susceptible prey (𝛼 ), the system will be convergent to the coexistence equilibrium From the Routh-Hurwitz criterion, we can conclude the following result. (𝐸 ). Increasing the value of 𝑎 or 𝛼 may cause the instability of 𝐸 and simultaneously may induce the stability of (𝐸 ). If Theorem 4. Equilibrium point 𝐸 is asymptotically stable the value of 𝑎 or 𝛼 is further increased, then equilibrium (𝐸 ) 2 3 ∗ ∗ 2 ∗ (locally) if 𝑐 𝑌 /(𝑘 +𝑆 ) <𝑟/𝐾 and <𝑐 . may be stable. u Th s, there exists a transcritical bifurcation 1 1 5 5 5 phenomenon driven by 𝑎 or 𝛼 . The dependence of stability Using similar arguments, the stability of coexistence properties of equilibrium can also be seen in Figure 2. It is equilibrium can be obtained as follows. seen that, for𝛼= 0 ,the coexistence point (𝐸 )isstable if 𝑎 ≤ 0.825. Furthermore, the free of infected prey equilibrium (𝐸 ) ∗ ∗ 2 Theorem 5. If 𝑐 𝑌 /(𝑘 +𝑆 ) <𝑟/𝐾 , then the coexistence 1 1 is stable for 0.825 < 𝑎 <1, while the prey free equilibrium point (𝐸 ) is asymptotically stable (locally). (𝐸 )is stable if 𝑎 ≥1.If we take 𝛼= 0.1 , then the 3 2 coexistence equilibrium, the free of infected prey, and the .. Global Stability. The main goal of pest control is to prey free equilibrium will be stable if 𝑎 ≤ 0.7333, 0.7333 < eradicate pest. Mathematically, this can be achieved whenever 𝑎 <0.9 and 𝑎 ≥0.9, respectively. 2 2 𝛽𝑆 푡㨀 푡㨀 𝑑𝑡 𝛽𝑆 𝑑𝑀 𝑑𝑡 𝑑𝐼 푡㨀 𝑟𝑆 푡㨀 𝑑𝑡 𝑟𝑆 𝑑𝑌 푡㨀 푡㨀 푡㨀 International Journal of Mathematics and Mathematical Sciences 5 0.5 0.4 0.3 0.2 0.1 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 a2 Figure 1: The stability regions of 𝐸 ,𝐸 and 𝐸 with respect to 𝑎 and 𝛼 . eTh parameter values are: 𝑟 = 1, 𝐾 = 1, 𝛽 = 0.1, 𝑐 = 0.01, 𝑐 = 3 5 2 1 0.5, 𝑐 =0.25,𝑘 =1,and 𝑘 =0.5. 2 1 2 0.6 0.1 0.1 2.5 S∗ S∗ Y 0.4 1.5 0.05 0.05 ∗ Y∗ S I∗ 0.2 ∗ ∗ ∗ ∗ I I S S 5 3 3 3 0 0 0.5 0 0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 1.2 ； ； ； ； 2 2 2 2 (a) (b) (c) (d) 0.4 2.5 2 5 1.5 0.2 Y∗ I∗ ∗ ∗ I I 5 3 0.5 0.4 0.6 0.8 1 1.2 0.4 0.6 0.8 1 1.2 ； ； 2 2 (e) (f) Figure 2: eTh stable equilibrium point as function of 𝑎 for ((a)-(c))𝛼= 0 and ((d)-(f))𝛼= 0.1 . eTh parameter values are 𝑟 = 1, 𝐾 = 1, 𝛽 = 0.1, 𝑐 = 0.01, 𝑐 =0.5,𝑐 =0.25,𝑘 =1,and 𝑘 =0.5. 1 2 1 2 Next we plot in Figure 3 the stability regions of 𝐸 ,𝐸 , the prey. In more detail, we also plot the dependence of 3 5 and 𝐸 using the same parameter values as before, except equilibrium points stability for 𝛼= 0 and 𝛼= 0.8 ;see 𝑎 =0.25 and varying the death rate of prey induced by Figure 4. It can be seen from Figures 4(a)–4(c), for 𝛼= 0 , the disease (𝑐 ). It is observed that, for relatively small rate of the coexistence point and the free of infected prey point will harvesting and 𝑐< 0.2 , the prey-free equilibrium is always be stable if 𝑐 < 0.0651 and 𝑐 ≥ 0.0651 , respectively. On the unstable. However, it can be seen that there is a minimal other hand, if we take 𝛼= 0.8 , then the prey free point will threshold of harvesting rate such that the prey-free state is be stable for any 𝑐 ; see Figures 4(d)–4(f). Hence system (1a), always stable while other equilibrium points are unstable. (1b), and (1c) also exhibits transcritical bifurcation driven by This threshold should be satisfied if we want to eradicate 𝑐 . 6 International Journal of Mathematics and Mathematical Sciences environment or may cause the crops poisonous. To minimize the cost for the pesticide control and also its side effects, Kar et al. [12]and Joshi [20]suggested to minimize the square of 0.8 the cost of applying the pesticide. Hence, our optimal control problem is to minimize the following objective functional: 0.6 (21) 𝐽= ∫ (+ 𝐼𝐵 + ) subject to system of differential equations (20a), (20b), and 0.4 (20c) and initial condition (2). Here, 𝐴 and 𝐵 are, respectively, the weight factors which correspond to susceptible and infected pest, while 𝐶 is the weight factor related to the use 0.2 of pesticide. The square of the control parameter is chosen to eliminate the side effects of the pesticide; see [12, 20]. Due to the practical limitations on the maximum rate of spraying pesticide, we consider the control set 0 0.05 0.10 0.15 0.20 Ω= {𝑢 (𝑡 ):0≤𝑢 (𝑡 )≤1,𝑡 ∈ [0, 𝑇 ]} . (22) We solve our optimal control problem using Pontryagin’s maximum principle. For that aim, we first define the Hamil- Figure 3: The stability regions of 𝐸 ,𝐸 and 𝐸 with respect to 𝑐 and 3 5 tonian 𝛼 . eTh parameter values are 𝑟 = 1, 𝐾 = 1, 𝛽 = 0.1, 𝑎 = 0.25, 𝑐 = 2 1 0.5, 𝑐 =0.25,𝑘 =1,and 𝑘 =0.5. 2 1 2 𝐻=𝐴𝑆+𝐼𝐵+𝑢𝐶 +𝜆 +𝜆 +𝜆 (23) 1 2 3 3. Optimal Control Problem where 𝜆 (𝑡), 𝑖 = 1, 2, 3are the adjoint (or costate) variables. The adjoint equations are We have discussed the dynamics of pest-predator model in which the pest population is controlled biologically by its natural enemy (predator) or the pest infection as well as 2𝑆 + 𝐼 =− =−𝐴− 𝜆 (𝑟 (1 − )−𝛽𝐼 mechanically by trapping (harvesting) the susceptible pest. In many cases, such control is not enough to eradicate the (24a) pest and therefore we need other types of controls. One of 𝑐 𝑘 𝑌 𝑐 𝑌 1 1 2 the well-known alternative controls is the use of pesticides. − −𝛼 −𝜖 𝑢) − 𝜆 − 𝜆 1 2 3 2 2 (𝑘 +𝑆) (𝑘 +𝑆) 1 2 The use of pesticide controls to reduce the population of pest has been studied theoretically in [10–12]. Here, we will also implement the pesticide as a pest control in addition to the =− =−𝐵+ 𝜆 ( +𝛽)𝑆 −𝜆 (𝛽𝑆 − 𝑐 1 2 (24b) previous biological and mechanical controls. For this aim, we modify system (1a), (1b), and (1c) by assuming that the −𝜖 𝑢) spraying of pesticide with rate 𝑢 reduces the growth rate of 𝑐 𝑆 2𝑐 𝑌 susceptible and infected prey, respectively, by the amount of 3 1 2 =− =𝜆 +𝜆 ( −𝑎 ), (24c) 1 3 2 𝜖 𝑆 and 𝜖 𝐼 ,where 𝜖 <𝜖 . Therefore, system (1a), (1b), and 𝑘 +𝑆 𝑘 +𝑆 1 2 1 2 1 2 (1c) is now modified as with transversality conditions 𝑆+ 𝐼 𝑐 𝑆𝑌 =𝑟(1− )𝑆 − 𝐼𝑆𝛽 − −𝛼𝑆 −𝜖 𝑆 (20a) 𝐾 𝑘 +𝑆 𝜆 (𝑇 )=𝜆 (𝑇 )=𝜆 (𝑇 )=0. (25) 1 2 3 (20b) =𝛽𝐼𝑆−𝑐𝐼−𝜖 𝐼 Using the optimality condition, 𝑐 𝑌 =(𝑎 − )𝑌 (20c) =0, (26) 𝑘 +𝑆 𝜕𝑢 subject to the initial conditions (2). the optimal control is characterized by Our main goal is to reduce the number of pest (suscep- tible and infected), by applying pesticide. However, we have ∗ ∗ 𝜆 𝜖 𝑆 +𝜆 𝜖 𝐼 ∗ 1 1 2 2 to consider the cost for applying pesticide which may be very (27) 𝑢 = . high as well as its side effects which may be harmful for the 2𝐶 𝑑𝑡 𝜕𝐻 𝑑𝑌 𝑑𝑡 𝑑𝐼 𝑑𝑡 𝑑𝑆 𝜕𝑌 𝑑𝑡 𝜕𝐻 𝑑𝜆 𝜕𝐼 𝑑𝑡 𝑟𝑆 𝜕𝐻 𝑑𝜆 𝛽𝐼 𝜕𝑆 𝑑𝑡 𝜕𝐻 𝑑𝜆 𝑑𝑡 𝑑𝑡 𝑑𝑡 𝑑𝑌 𝑑𝐼 𝑑𝑆 𝑑𝑡 𝐶𝑢 𝐴𝑆 International Journal of Mathematics and Mathematical Sciences 7 0.5 1.5 0.75 0.75 0.5 0.5 Y 0.25 Y∗ S∗ 0.5 0.25 0.25 I∗ 5 S 0 0 0 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 0 0.1 0.2 c c c c (a) (b) (c) (d) 0.5 1 0.25 0.5 0 0 0 0.1 0.2 0 0.1 0.2 c c (e) (f) Figure 4: eTh stable equilibrium point as function of 𝑐 for ((a)-(c))𝛼= 0 and ((d)-(f)) 𝛼= 0.8 . eTh parameter values are 𝑟 = 1, 𝐾 = 1, 𝛽 = 0.1, 𝑎 = 0.25, 𝑐 =0.5,𝑐 = 0.25, 𝑘 =1,and 𝑘 =0.5. 2 1 2 1 2 Since the control variable has to be in Ω, the characterization perform a simulation with initial values 𝑆(0) = 0.3, 𝐼(0) = 0.1 is then given by and 𝑌(0) = 0.2, where the pesticide control is assumed to be applied in 100 units of time; i.e., 𝑇 = 100 .The number of susceptible pest, infected pest, and predator as function of time, both without and with pesticide control, are plotted in ∗ ∗ 0 if 𝜆 𝜖 𝑆 +𝜆 𝜖 𝐼 <0 1 1 2 2 Figures 4(a)–4(c). It is observed that without control, both (28) ∗ ∗ 𝜆 𝜖 𝑆 +𝜆 𝜖 𝐼 pest and predator survive in the environment. However, the 1 1 2 2 ∗ ∗ if 0≤𝜆 𝜖 𝑆 +𝜆 𝜖 𝐼 ≤2𝐶 1 1 2 2 optimal application of pesticide control can eradicate both { 2𝐶 ∗ ∗ susceptible and infected pest. Since the growth of predator 1 if 𝜆 𝜖 𝑆 +𝜆 𝜖 𝐼 >2.𝐶 { 1 1 2 2 also depends on the number of susceptible pest population, We now have our optimality system consisting of the state the population of predator is also reduced. In other words, the equations (20a), (20b), and (20c) with initial conditions (2), use of pesticide aec ff ts not only the pest population but also the adjoint (costate) equations (24a), (24b), and (24c) subject the population of predator. Furthermore, we also plot optimal to the transversality conditions (25), and the optimal control pesticide control in Figure 5(d). From this gur fi e we see that (28). This optimality system will be solved numerically the use of pesticide is optimal when it is applied at maximum using the forward-backward sweep method [21]. We start level for about 50 units of time and then reduce gradually the by taking an initial guess of 𝑢(𝑡). Using the forward fourth- pesticide control until it stops at final time. order Runge-Kutta method, we solve the state-equations with its initial conditions. Using the solutions of these state 4. Conclusion variables, we successively solve the adjoint equations with the transversality conditions by the backward fourth-order We have proposed an ecoepidemic pest-predator model Runge-Kutta method. The control variable is then updated by describing the interaction of pest with its natural enemy substituting the values of the state and adjoint solutions into where the pest may be infected by pathogens (bacteria, fungi, (28). This process is repeated until a convergent solution is or viruses). We also consider the effect of pest harvesting. achieved. The presence of natural enemy and infectious pathogens is Since we are not considering a specific case of quantitative considered as biological control for the pest population, while nature, we use the hypothetical parameter values as 𝑟= pest harvesting can be considered as one of mechanical pest 1,𝐾 = 1, 𝛽 = 0.02, 𝑐 = 0.01, 𝑐 =0.3,𝑐 =0.2,𝑘 =1,𝑘 = control. Our model is based on the Leslie-Gower equation in 1 2 1 2 0.75, 𝑎 =0.5,and𝛼= 0.2 . In this case, system (1a), (1b), and which the predator (natural enemy) grows logistically where (1c) is convergent to the free of infected pest equilibrium 𝐸 = its carrying capacity depends on the number of susceptible ∗ ∗ 2 (0.205,0.0, 2.388) because 𝑐 𝑌 /(𝑘 +𝑆 ) = 0.4932 < /𝐾𝑟 = pest and other natural resources for the predator. us, Th 1 5 1 5 1, meaning that the pest will always be endemic. To reduce the although there is no pest, the predator still survives due to number of pest we apply pesticide control. To minimize the other natural resources. This explains why the extinction total population of pest, we set the weight factors𝐴=𝐵 = 1 . of both pest and predator equilibrium (𝐸 ), the survival of By keeping in mind that the cost to kill a single pest is quite susceptible prey equilibrium (𝐸 ), and the predator free equi- low, we take 𝐶 = 0.025. Using those values of parameter we librium (𝐸 ) cannot be stable. The primary aim of controlling 4 8 International Journal of Mathematics and Mathematical Sciences 0.1 0.6 0.4 0.05 0.2 0 20 40 60 80 100 0 20 40 60 80 100 without control without control with control with control (a) (b) 3 1 0.5 0 20 40 60 80 100 0 20 40 60 80 100 t t without control without pesticide, u = 0 with control optimum control of pesticide, u∗ (c) (d) Figure 5: The curves of state variables for system (20a), (20b), and (20c) without and with control: (a) susceptible pest, (b) infected pest, (c) predator, and (d) optimal control variable. pest is to eliminate the pest from agriculture. u Th s, the pest Conflicts of Interest free equilibrium (𝐸 ) is the most important equilibrium due The authors declare that there are no conflicts of interest to the fact that if this equilibrium is stable then the pest is regarding the publication of this paper. totally eradicated. 𝐸 will be globally asymptotically stable if 𝑟< 𝛼 +𝑐 𝑎 𝑘 /𝑐 (𝑘 +𝐾) . It is observed that the stability of 1 2 2 2 1 Acknowledgments E may be achieved if the rate of pest harvesting is suitably high. This work was supported by the Directorate of Research and In some cases, the combination of biological and mechan- Community Service, The Directorate General of Strengthen- ical control to eliminate the pest may be not very effective ing Research and Development, and the Ministry of Research, and therefore we need to include chemical control such as Technology and Higher Education (Brawijaya University), the use of pesticide. To consider the effect of such chemical Indonesia, Contract no. 137/SP2H/LT/DRPM/III/2016 dated control, we modify our model by including the effect of March 10, 2016, and Contract no. 460.18/UN10.C10/PN/2017 the use of pesticide. Then we formulate the optimal control dated April 18, 2017. problem to minimize the total number of pest, the cost of applying pesticide, and also the side effect of pesticide. 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Published: Jun 18, 2019
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