PERSISTENCE AND EXTINCTION OF SINGLE POPULATION IN A POLLUTED ENVIRONMENT

Transcription

1 Electronic Journal o Dierential Equations, Vol 2004(2004), No 108, pp 1 5 ISSN: URL: or tp ejdemathtxstateedu (login: tp) PERSISTENCE AND EXTINCTION OF SINGLE POPULATION IN A POLLUTED ENVIRONMENT ZHAN LI, ZHISHENG SHUAI, KE WANG Abstract In this paper, we consider the ODE system corresponding to a diusive-convective model or the dynamics o a population living in a polluted environment Suicient criteria on persistence and extinction o the population are derived 1 Introduction Today, the most threatening problem to the society is the change in environment caused by pollution, aecting the long term survival o species, human lie style and biodiversity o the habitat Thereore the study o the eects o toxicant on the population and the assessment o the risk to populations is becoming more important In the early eighties a deterministic modelling approach to the problem o assessing the eects o a pollutant on an ecological system was proposed by Hallam and his co-workers[4, 5, 6] Since then, such models have been the subject o many investigations and improvements Population-toxicant coupling has been applied in several contexts, including Lotka-Volterra and chemostat-like environments, resulting in ordinary, integro-dierential and stochastic models Usually a qualitative analysis was perormed which ocuses on the survival or extinction o populations [9, 10] All these studies rely on the hypothesis o a complete spatially homogeneous environment Recently, a irst attempt to consider a spatial structure has been carried out in [2, 3] where a reaction-diusion model is proposed to describe the dynamics o a living population interacting with a toxicant present in the environment(external toxicant) through the amount o toxicant stored into the bodies o the living organisms(internal toxicant) However, as the authors pointed out, even i the resulting model presents many eatures which make stimulating its study, such a modelling approach is a rough approximation to the biological phenomena at hand In 1999, Buonomo etal viewed the internal toxicant as drited by the living population and then, by balance arguments, they derived a PDE system consisting into two reaction diusion equations coupled with a irst order convection equation, and the 2000 Mathematics Subject Classiication 93C15 Key words and phrases Population dynamics; environment pollution; persistence c 2004 Texas State University - San Marcos Submitted January 9, 2004 Published Sept 12, 2004 Supported by the National Natural Science Foundation o China (No ), and by the Key Project on Science and Technology o the Education Ministry o China (No Key 01061) 1

2 2 Z LI, Z SHUAI, K WANG EJDE-2004/108 corresponding ODE system was obtained as well [1] This model is the most realistic by now but the analysis o it is so diicult that they only used some analytic and numerical approaches Obviously, the more clear work is deserved to do In this paper, we use some new methods to investigate the model made by Buonomo et al and the conditions o survival and extinction are obtained 2 The Model We utilize a modiied logistic equation [8] to model the eect o toxin on single species We take n(t): concentration o the population biomass c(t): concentration o the toxicant in the environment z(t): concentration o the toxicant in the population We assume that there is a given(external) toxicant in the environment, and the living organisms absorb into their bodies part o this toxicant so that the dynamics o the population is aected by this(internal) toxicant Concerning the growth rate o the population we assume that the birth rate is b(n) = b 0 n and the death rate is d(n, c) = d 0 + αc, where b 0,d 0 and α are positive constants is assumed to be a non-negative constant Thereore we assume, in absence o toxicant, a malthusian( = 0) or a logistic growth rate( > 0) We can see that i b 0 d 0 αc 0, n(t) will be extinct in the end, so we suppose c < b 0 d 0 := c 1 (21) α We propose the ollowing model governing the system with initial data dn dt = n(b 0 d 0 αc n) dc dt = kz (r + m + b 0 n)c dz dt = kzn + (r + d 0 + αc)cn hz + u(t) n(0) = n 0 0; c(0) = c 0 0; z(0) = z 0 0 (22) Here α is the depletion rate coeicient o the population due to organismal pollutant concentration k is the depletion rate o toxicant in the environment due to its intake made by the population r is the depletion rate o toxicant in the population due to egestion m is the depletion rate o toxicant in the population due to metabolization processes h is the depletion rate o the toxicant in the environment u is the exogen+ous toxicant input rate which is assumed to be a smooth bounded non-negative unction o t 3 The conditions o survival and extinction or the population when u(t) = Q > 0 We now recall the deinitions o persistence and extinction A component n(t) o a given ODE system is said to be persistent i or any n(0) > 0 it ollows that n(t) > 0, t > 0 and lim in n(t) > 0 I there exists δ > 0 (independent o n(0))

3 EJDE-2004/108 PERSISTENCE AND EXTINCTION 3 such that n(t) is persistent and is bounded and lim in n(t) δ, then n(t) is said to be uniormly strongly persistent I there exists δ > 0 (independent o n(0)) such that n(t) δ, then n(t) is said to be uniormly weakly persistent I n(t) 0, then n(t) is said to be extinct Theorem 31 The system (22) is uniormly strongly persistent i and only i αkq < h(b 0 d 0 )(r + m + b 0 ) Proo First, we deduce that the system is uniormly weakly persistent Assume that the system is not uniormly weakly persistent then there exists a sequence o initial value (n k (0), c k (0), z k (0)) (0, + ) R 2 +, such that n k (t) = ε k 0, Then there exist T k > 0, such that From (21) and (31) we have as k + n k (t) < 2ε k or t T k (31) z k (r + d 0 + αc)cn k hz k + Q (r + b 0 )c 1 2ε k hz k + Q or t T k Using the Comparison Theorem we have z k (t) c 1(r + b 0 )2ε k + Q h So or all ε > 0, there exists H k > T k > 0, such that z k (t) c 1(r + b 0 )2ε k + Q + ε =: z k h or t H k (32) From (31) and (32) we see that c k kz k (r + m + b 0 )c k + 2ε k c k = kz k (r + m + b 0 2ε k )c k or t H k Similarly by the Comparison Theorem and let ε 0, we have Then, or t H k, c k (t) kc 1(r + b 0 )2ε k + kq h(r + m + b 0 2ε k ) c k (t) kc 1(r + b 0 )2ε k + kq h(r + m + b 0 2ε k ) := β(ε k) (33) Now we consider the irst equation o the model (22), rom (33) it is easy to see there exists S k > H k > 0, such that ṅ k n k (b 0 d 0 αβ(ε k ) n k ) or t S k Using the Comparison Theorem again we have lim in n k(t) b 0 d 0 αβ(ε k ) (34) By (34) and the assumption n k (t) = ε k, we obtain n k (t) = ε k lim in n k(t) b 0 d 0 αβ(ε k )

4 4 Z LI, Z SHUAI, K WANG EJDE-2004/108 Let k +, it ollows that ε k 0 and β(ε k ) 0 b 0 d 0 αβ(ε k ) kq h(r+m+b 0) Hence, h(b 0 d 0 )(r + m + b 0 ) αkq h(r + m + b 0 ) That is, αkq h(b 0 d 0 )(r + m + b 0 ) So there is uniormly weakly persistent i αkq < h(b 0 d 0 )(r + m + b 0 ) Using the well known result which says that uniorm weak persistence implies uniorm strong persistence [7, Section 2], then the proo is completed Moreover, i we look at the system restricted to {0} R 2 +, then there is a unique equilibrium X = (0, c, z) with z = Q h and c = kz (r + m + b 0 ) Then an easy investigation o the linearized equation at X shows that when αkq > h(b 0 d 0 )(r + m + b 0 ), X is locally asymptotically stable In particular the system is not uniormly persistent anymore Theorem 32 Consider the system (22) I Q > (r+m+b0)(b0 d0)(h+k(b0 d0)) αk, then the population is extinct Proo From (31) we know n(t) n 1, or t > t 1 So rom the last equation o the model (22), we can obtain dz dt > Q hz kn 1z or t > t 1 Now we use the Comparison Theorem again, then we have lim in z(t) Q h + kn 1 Let ε 0, we have Q lim in z(t) = h + k(b 0 d 0 ) := m z That is to say or all ε > 0, T 1, such that z(t) > m z ε, or all t > T 1 From the second equation o the model (22), i t > T 1, it is easy to see dc dt > k(m z ε) (r + m + b 0 )c Similarly by the Comparison Theorem, Let ε 0 Then we have lim in c(t) lim in c(t) km z ε) r + m + b 0 k(m z r + m + b 0 = kq (r + m + b 0 )(h + k(b 0 d 0 )) =: m c Then or all ε > 0 there exists T 2 > T 1, such that c(t) > m c ε or all t > T 2 Obviously, rom the irst equation o the model (22), i t > T 2, we have dn dt < n(b 0 d 0 α(m c ε) n)

5 EJDE-2004/108 PERSISTENCE AND EXTINCTION 5 By the Comparison Theorem, we have Let ε 0 Then we obtain Clearly, n(t) b 0 d 0 α(m c ε) n(t) b 0 d 0 αm c =: M n n(t) M n b 0 d 0 I M n < 0, that is Q > (r+m+b0)(b0 d0)(h+k(b0 d0)) αk, the population will be extinct Acknowledgements We thank the anonymous reeree or his/her remarks that helped us improving Theorem 31 Reerences [1] B Buonomo, A D Liddo, I Sgura, A diusive-convective model or the dynamics o population-toxocant intentions: some analytical and numerical results, Math Biosci 157(1999), [2] B Buonomo, A D Liddo, On population dynamics in a diusive polluted environment: a single species model, Studi Urbinati 1(1997), 69 [3] B Buonomo, A D Liddo, Large time behaviour o a population-toxicant interaction model, RappInt IRMA-CNR 4/ [4] TG Hallam, C E Clark, R R Lassiter; Eects o toxicants on populations: A qualitative approach I Equilibrium environment exposure, Ecol Model 18(1983), [5] T G Hallam, C E Clark, G S Jordan; Eects o toxicants on populations: A qualitative approach II First Order Kineics, J Math Biol 18(1983), [6] T G Hallam, J T Deluua; Eects o toxicants on populations: A qualitative approach III Environmental and ood chain pathways, JTheoretBiol 109(1984), [7] Horst R Thieme, Uniorm persistence and permanence or non-autonomous semilows in population biology, Math Biosci 166(2000), [8] H I Freedman, Deterministic Mathematical Models in Population Ecology, Marcel Dekker, New York, 1980 [9] Debasis Mukherjee, Persistence and global stability o a population in a polluted environment with delay, Journal o Biological Systems 10(2002), [10] Z Ma, G Cui, WWang, Persistence and extinction o a population in a polluted environment, Math Biosci 101(1990), 75 Zhan Li Department o Mathematics, Key Laboratory or Vegetation Ecology o Education Ministry, Northeast Normal University, Changchun , China address: lizhan1125@hotmailcom Tel: Zhisheng Shuai Department o Mathematics, Key Laboratory or Vegetation Ecology o Education Ministry, Northeast Normal University, Changchun , China address: shuaizs@hotmailcom Ke Wang Department o Mathematics, Harbin Institute o Technology, Weihai , Shandong, China address: w k@hotmailcom