污染环境下具有Markov切换的随机食饵-捕食渔业模型及其最优收获.pdf

应用数学MATHEMATICA APPLICATA2024,37(2):303-320Optimal Harvesting for a StochasticPrey-Predator Fishery Model withMarkovian Switching in a PollutedEnvironmentZHONG Qiqi(钟琪琪)1,WEI Yuming(韦煜明)2(1.College of Intelligent Medicine and Biotechnology,Guilin MedicalUniversity,Guilin 541199,China;2.School of Mathematics and Statistics,Guangxi Normal University,Guilin 541006,China)Abstract:This article investigates the optimal harvesting strategy of a stochasticprey-predator fishery model with white noise and Markovian switching in a pollutedenvironment.The existence and uniqueness of the positive global solution are first shown.It discusses thresholds for population extinction and persistence.The sufficient conditionfor the existence of an ergodic stationary distribution is derived by constructing anappropriate Lyapunov function.The ergodic method is applied to study the optimalharvesting strategy of the model.Finally,some numerical simulations are provided todemonstrate the theoretical analysis.The analysis shows that Markovian switching couldsuppress the extinction of species,while white noise intensity and pollutant concentrationhave a significant negative effect on optimal harvesting.Key words:Prey-Predator model;Optimal harvesting;Environmental pollution;Markovian switchingCLC Number:O175AMS(2010)Subject Classification:91B76;92D25;60H10Document code:AArticle ID:1001-9847(2024)02-0303-181.IntroductionUnexpected environmental shocks and pollutant releases are known to impact fish pop-ulation dynamics.One of the current hot issues is the negative consequences of pollutiondischarges on fisheries and mariculture.Estimating environmental toxins and developingmatching mathematical models to generate more effective harvesting tactics are required toavoid economic losses in fisheries and safeguard ecosystem balance.Pioneered by Clark1,the bioeconomic models have become an advantageous tool forstudying optimal harvesting strategies of renewable resources such as fishery resources.Forexample,Das et al.2first proposed a venomous prey-predator fishery model in which preyReceived date:2023-03-02Foundation item:Supported by the National Natural Science Foundation of China(11961074)Biography:ZHONG Qiqi,female,Han,Guangxi,major in biomathematics.Corresponding author:WEI Yuming.304MATHEMATICA APPLICATA2024populations grow logically,the predator mortality index decreases,and functional responsesare proportional to prey numbers.Ang et al.3discussed the optimal harvest fishery model,which is influenced by environmental toxicity and independent fishing strategies,and so on.In reality,when the contaminant emissions are significant,the population will graduallydecrease,even becoming extinct.Due to the increasing influence of random factors,thedeterministic models are challenging to describe the systems state accurately.Moreover,the obtained threshold will vary significantly from the actual situation.In other words,theunexpected environmental changes in ecosystems unavoidably affect population growth,andsuch changes are an essential aspect of population evolution.Therefore,many researchers havediscussed the dynamics of stochastic population systems by simulating minor environmentalfluctuations with white noise46.WU and WEI4supposed the contact rate between adultsand toxicants in the environment which is disturbed by white noise,then investigated asingle-species population model in polluted environments with psychological effects and partialtolerance.YANG et al.5analyzed a population model in which the intrinsic growth rate isperturbed by linear functions of Gauss white noise,then gived the unique optimal harvestingeffort and a corresponding expectation of maximum sustainable yield.NING et al.6studiedpopulation models with saturation effect and impulsive toxicant input,and used a mean-reverting Ornstein-Uhlenbeck process to model the random changes in the growth rate whichwas produced by randomly varying environment.They further analyzed the conditions underwhich two species permanently survive or move towards extinction.On the other hand,telegraph noise may also impact the population system.This noise causes the system toswitch from one environmental state to another,which is memoryless,with the duration ofeach environmental state obeying an exponential distribution.A continuous-time Markovchain can be used to represent this switching78.The deterministic models in 2-3 all assumed that species were affected by specific toxicitycoefficients and did not take into account the loss and uptake of pollutants.Moreover,theoptimal control strategy was obtained by using Pontryagins maximum principle in 3,5,whilethe associated optimal harvesting policy was not studied in 4,6,9-10.Stochastic systems incontaminated environments were proposed in 4,6,10,but the authors did not discuss theoptimal harvesting strategies.In contrast,optimal harvesting strategies of some stochasticsystems were investigated in 5,but environmental pollution was not taken into account.Based on the discussion above,we build a more realistic stochastic logistic predator-preyfishery model that considers Markovian switching,as well as contaminant concentrations in theorganism and the external environment.Inspired by 2,we postulate that prey and predatorfish species are subject to be harvested with harvesting effort of q1 0 and q2 0,respectively.This study aims to find the optimal harvesting effort that maximizes the expectation ofsustainable yields without leading to the extinction of any species.The structure of this article is as follows.The proposed stochastic logistic fishing modelis described in the next section,along with associated notations,lemmas,and assumptions.A threshold theorem for determining species extinction or persistence in the mean will begiven and proven in Sect.3.In Sect.4,we present the results of an approximation to theoptimal harvesting policy,and proves the existence and uniqueness of the invariant probabilitymeasure.Carrying out some numerical simulations to demonstrate theoretical results in Sect.No.2ZHONG Qiqi,et al.：Optimal Harvesting for a Stochastic Prey-Predator Fishery Model3055.We conclude the work with a brief discussion and some recommendations for future researchin Sect.6.2.The ModelThe n different environmental state space are denoted as S=1,2,n.These statesare obtained by dividing them according to temperature,humidity,light,etc.Let(t),t 0be a continuous-time Markov chain whose values are taken from a finite state space S.TheMarkov chain(t),t 0 is generated by=(ij)nn,i.e.,P(t+t)=k|(t)=i)=ikt+o(t),if i=k,1+iit+o(t),if i=k,where t 0,ikdenotes the transfer rate from i to k.If i=k,then there is ik 0 whileii=i=kikfor each i=1,2,3,n.We extend the main work in 3,and obtain the following stochastic fishery model withMarkovian switching in a polluted environment:dx1(t)=x1(t)r1(1 x1(t)K1)1x2(t)q1 1(t)1C1(t)dt+x1(t)1(t)dW1(t),dx2(t)=x2(t)r2(1 x2(t)K2)+2x1(t)q2 2(t)2C2(t)dt+x2(t)2(t)dW2(t),dC1(t)dt=u1Ce(t)(l1+d1)C1(t),dC2(t)dt=u2Ce(t)(l2+d2)C2(t),dCe(t)dt=u(t)mCe(t),(2.1)where the density of the ith species at time t is represented by xi(t)(i=1,2).u(t)is theexogenous total toxicant input into the environment at time t,and Ce(t)is the toxicantconcentration in the environment at time t.The toxin concentration of xi(t)in the organismat time t are denoted as Ci(t)(i=1,2).1,2represent the predator rate and nutritionalconversion rate,respectively.i(k)(i=1,2)represent white noise intensity in regime k S.Wi(t)(i=1,2)are the mutually independent Brownian motions.Suppose that(),W()areindependent of each other in the same complete probability space(,F,Ftt0,P).TheMarkov chain(t),t 0 is assumed to be irreducible.The unique stationary distributio=(1,2,n)of(t),t 0 can be obtained by solving the following equation=0,ni=1i=1,where i 0,i S.Positive values are chosen for all parameters in the model(2.1),andthe description of the other parameters are listed in Tab.2.1.306MATHEMATICA APPLICATA2024Tab.2.1The biological significance of the parameters in the model(2.1)ParametersBiological significancerithe intrinsic growth rate of ith fish speciesKithe carrying capacity of xi(t)ithe mortality rate of xi(t)ithe dose-response rate of xi(t)lithe net ingestion rate of xi(t)mthe contaminants loss rate from environmentuithe toxin net uptake rate of xi(t)dithe deportation rate of toxicant Ci(t)Noting C1(t),C2(t)and Ce(t)can be solved explicitly,then only the first two equationsin the system(2.1)need to be considered as follows:dx1(t)=x1(t)r1(1 x1(t)K1)1x2(t)q1 1(t)1C1(t)dt+x1(t)1(t)dW1(t),dx2(t)=x2(t)r2(1 x2(t)K2)+2x1(t)q2 2(t)2C2(t)dt+x2(t)2(t)dW2(t),(2.2)with initial valuesx1(0)=x10 0,x2(0)=x20 0,(0)S.We define the following notations for the sake of ease and simplicity:i(t)=0.52i(t)+i(t),i=nk=1ki(k),i=1,2.bi=ri qii iCi,i=1,2.x(t)=1tt0 x(s)ds,x=limsupt+x(t),x=liminft+x(t),=12+r1r2K1K2,1=?b1b21r2K2?,2=?2b2r1K1b1?.Before elaborating on our results,let us introduce the following theorem,lemmas,andassumptions.Lemma 2.110Let x(t)C(0,+,R+),then(i)If there are positive constants T,0,such thatlnx(t)t 0t0 x(s)ds+2i=1iWi(t)a.s.,for all t T,where iare constants,thenx(t)0,a.s.,if 0;limt+x(t)=0,a.s.,if 0,0,0 0,such thatlnx(t)t 0t0 x(s)ds+2i=1iWi(t),a.s.,for all t T,where iare constants,then x(t)0,a.s.Assumption 2.1Suppose thatlimt+u(t)=u is true.No.2ZHONG Qiqi,et al.：Optimal Harvesting for a Stochastic Prey-Predator Fishery Model307Lemma 2.29If Assumption 2.1 is valid,thenlimt+Ce(t)=um,limt+1tt0Ci(s)ds=ui um(li+di),Ci,i=1,2.C1(t),C2(t)and Ce(t)are the pollutant concentrations,then 0 C1(t)1,0 C2(t)1,and 0 Ce(t)1 must be true for all t 0.Theorem 2.1If Lemma 2.1 hold,then for any given initial value(x1(0),x2(0),(0)R2+S,the model(2.1)has a unique solution x(t)=(x1(t),x2(t)R2+,a.s.ProofClearly,the coefficients of the model(2.1)are local Lipschitz continuous,hencethere exists a unique local solution(x1(t),x2(t)on t 0,e)by the theorem about existenceand uniqueness for stochastic differential equations,and edenotes explosion time11.weneed to show(x1(t),x2(t)is the global solution,i.e.,e=+,a.s.Set n0is large enoughsuch that1n0 xi(0)n0,i=1,2.For each n n0,define the stopping timen=inft 0,e):xi(t)/(1n0,n0),for i=1,2,where inf =+(denotes the empty set).It is easy to see that nincreases as n tends toinfinity.Let+=limn+n,where+e,a.s.It follows from+=+,a.s.thate=+,a.s.,therefore(x1(t),x2(t)R2+almost surely for all t 0.Defined a C2-functionV1:R2+R+asV1(x1(t),x2(t)=2i=1xi(t)lnxi(t)1.Let n n0,and T 0 for arbitrary 0 t n T.Using generalized It os formula11toV1(x1(t),x2(t)yieldsdV1(x(t)=LV1(x(t)+2i=1i(k)(xi(t)1)dWi(t),whereLV1(x(t)=2i=1(xi(t)1)ri(1 riKi)qi i(k)iCi(t)2i,j=1,i=j(1)jixj(t)+1221(k)+1222(k),r1K1x21(t)r2K2x22(t)+2x1(t)x2(t)+(r1+r1K1)x1(t)+(2+r2+r2K2)x2(t)+2i=1qi+i(k)+iCi(t)+122i(k).There exists a positive constant c such thatLV1(x(t)c +,x(t)R2+.(2.3)Integrating(2.3)from 0 to n T on both sides and taking the expectation,we getEV1(x(n T)V1(x(0)=EnT0LV1(x(s)ds cE(n T)cT.ThenV1(x(0)+cT EV1(x(n T)EV1(x(n)In,308MATHEMATICA APPLICATA2024where n=n|n T,Inis the indicator function of n T.According to thedefinitions of V1(x(t)and n,we can obtainV1(x(n)(n 1 lnn)(1n 1 ln1n),hencecT+V1(x(0)(1n 1 ln1n)(n 1 lnn)P(n).Letting n tends to+,thenlimn+P(n)=0.Due to the arbitrariness of T,one can getP+=+=1,thus+=+,a.s.Lemma 2.3Under the conditions of Theorem 2.1,there exists positive constantsCi(p),p such thatlimsupt+Expi(t)Ci(p),i=1,2,andlimsupt+lnxi(t)lnt 1,a.s.,i=1,2.ProofApplying It os formula to Lyapunov function etV2(x(t)=et2i=1xpi(t)yieldsd(etV2(x(t)=etV2(x(t)dt+dV2(x(t)=et2i=1xpi(t)(1+p(ri(1 xi(t)Ki)qi iCi(t)i(k)2i,j=1,i=j(1)jixj(t)+12p(p 1)2i(k)dt+pxpi(t)i(k)dWi(t),etL(V2(x(t)dt+2i=1pxpi(t)i(k)dWi(t),(2.4)whereL(V2(x(t)=2i=1xpi(t)(1+p(ri(1 xi(t)Ki)qi iCi(t)i(k)2i,j=1,i=j(1)jixj(t)+12p(p 1)2i(k)2i=1xpi(t)(1+pri+0.5p22i(k)+2pxp2(t)x1(t)Ci(p)+.Integrating both sides of(2.4)from 0 to t and taking the expectation yieldsetEV2(x(t)=etEV2(x(0)+Et0esL(V2(x(s)ds 0 such thatE(supuu+1V3(x()C5,u=1,2,.For arbitrary 0,then by virtue of Chebyshevs inequality11,we haveP:suputu+1V3(x(t)u1+C5u1+,u=1,2,.According to Borel-Canellis lemma11,there exists a u0such that for almost all ,ifu u0and u t u+1,suputu+1V3(x(t)u1+.Thuslnxi(t)lntsuputu+1V3(x(t)lntlnu1+lnt 1+,i=1,2,a.s.By the arbitrariness of,we havelimsupt+lnxi(t)lnt 1,a.s.,i=1,2.The proof is completed.3.Persistence and ExtinctionWe will calculate the persistence and extinction thresholds for fish populations in thissection.310MATHEMATICA APPLICATA2024Assumption 3.1qi ri i(t)0.52i(t)iCi,i=1,2.According to Assumption 3.1,fish populations can exist alone in a contaminated envi-ronment if white noise and fishing efforts are small enough.If Assumption 3.1 is violated,allspecies will become extinct,and the model(2.1)will no longer be worth discussing.Theorem 3.1For the model(2.2),suppose Assumption 3.1 holds.(i)If b1 0,b2 0,then both species are extinct almost surely.(ii)If b1 0,then the species x1(t)will be extinct almost surely,and the speciesx2(t)will almost surely persist in the mean,i.e.,limt+x1(t)=0,limt+1tt0 x2(s)ds=b2K2r2,a.s.(iii)If b1 0,b2 0,b2 0,and r2b1 0,b2 0,and r2b1 1b2K2,then both the species x1(t)and x2(t)are stablein time average almost surely,i.e.,limt+1tt0 xi(s)ds=i,i=1,2.ProofApplying generalized It os formula to lnxi(t)(i=1,2),and integrating bothsides from 0 to t gives1tlnx1(t)x1(0)=r1 q1 1C1(t)r1K1x1(t)1x2(t)1tt01(s)ds+1tt01(s)dW1(s),(3.1)and1tlnx2(t)x2(0)=r2 q2 2C2(t)r2K2x2(t)+2x1(t)1tt02(s)ds+1tt02(s)dW2(s).(3.2)Utilizing the strong law of large numbers for local martingales12,we havelimt+1tt0i(s)dWi(s)=0,a.s.,i=1,2.(3.3)In light of Lemma 2.2 and the ergodicity of(),one obtain thatlimt+Ci(t)=Ci,limt+t1t0i(s)ds=i,a.s.,i=1,2.(3.4)(i)Making use of(3.1)and(3.4),for any arbitrary,there exists a T 0 such thatC1(t)C121,1tt01(s)ds 12for all t T,1tlnx1(t)x1(0)b1r1K1x1(t)+1tt01(s)dW1(s).(3.5)No.2ZHONG Qiqi,et al.：Optimal Harvesting for a Stochastic Prey-Predator Fishery Model311Combining b1 0,there exists a T 0 suchthat C2(t)C232,1tt02(s)ds 23,and 2x1(t)3holds for all t T.Thus1tlnx2(t)x2(0)b2+r2K2x2(t)+1tt02(s)dW2(s).By the arbitrariness of and Lemma 2.1,it follows thatlimt+x2(t)=0,a.s.(ii)If b1 0,there exists a T 0 such that for all t T,3 2x1(t)3,C232 C2(t)C2+32,23 2(t)2+3.(3.6)Substituting(3.6)into(3.2),one can obtain1tlnx2(t)x2(0)b2 r2K2x2(t)+1tt02(s)dW2(s),(3.7)and1tlnx2(t)x2(0)b2+r2K2x2(t)+1tt02(s)dW2(s).(3.8)Combining Lemma 2.1,(3.3),(3.7)and(3.8),one can deduce that(b2)K2r2 x2(t)x2(t)(b2+)K2r2,a.s.Then by the arbitrariness of,one can abtainlimt+x2(t)=b2K2r2,a.s.(iii)The proof of(iii)is the same as(ii),thus it is omitted here.(iv)Computing(3.1)r2K2(3.2)1yieldsr2K2tlnx1(t)x1(0)1tlnx2(t)x2(0)=r2K2(r11tt01(s)ds q1 1C1(t)x1(t)1(r21tt02(s)ds q2 2C2(t)+r2K21tt01(s)dW1(s)1t1t02(s)dW2(s).(3.9)According to Lemma 2.2 and(3.4),for any 0,there exists a T 0 such that1tlnx2(t)x2(0)+r2K2lnx1(0)3,r21K2C1(t)12C2(t)r21K2C1 12C23,r2K2t1t01(s)ds 1t1t02(s)ds r2K21 123.Substituting these inequalit