Resilience and stability of harvested predator-prey systems to infectious diseases in the predator

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1 Resilience and stability of harvested predator-prey systems to infectious diseases in the predator Morgane Chevé Ronan Congar Papa A. Diop November 1, 2010 Abstract In the context of global change, emerging risks associated with parasites, invasive species and infectious diseases are an important issue especially for developing countries whose economies are resource-based. Our objective is to provide a mathematical framework to study the response of a harvested predator-prey system to a disease in the predator and proportional harvesting of predator. We combine a classical predator-prey model with an eco-epidemiological model where predator disease is modeled by a Susceptible-Infected (SI) epidemic system. Our results question the adoption of conservative harvest policies in order to enhance the resilience and stability of harvested ecological systems to emerging infectious diseases. Indeed we show that harvest can be an effective strategy for controling the spread of the disease and enhancing the resilience of the ecosystem. Whereas the disease can become endemic in the population at low harvest rates, it can be completely eradicated by harvesting the predator at a sufficiently high rate. Keywords: eco-epidemiological model, harvesting, predator prey system, threshold, stability. JEL classification: C61, Q20, Q22. 1 Introduction In the context of global change, emerging risks associated with parasites, invasive species and infectious diseases caused by pathogens (such as viruses and bacteria) are an important issue for biodiversity and ecosystems conservation. In their seminal paper Anderson and May [1] proposed an eco-epidemiological model to deal with infectious diseases in predator-prey systems. Their approach couples the epidemiological model developed by Université du Havre et CARE (morgane.cheve@univ-rouen.fr). Université de Rouen et CARE (ronan.congar@univ-rouen.fr). Université de Rouen et CARE, 3 avenue Pasteur, Rouen Cedex 1, France (papa.diop@univ-rouen.fr). 1

2 Kermack and McKendrick to a classical Lotka and Volterra predator-prey model. From this work, many authors have studied the effects of a disease in the prey on the dynamics of the predator-prey system (Chattopadhyay and Arino [7], Chattopadhyay and Bairagi [8], Hethcote et al. [17] and Xiao and Chen [20]) or considered the case where the predator only is affected by a disease (Han et al. [15], Haque and Venturino [16], Venturino [18]). Some authors also consider the case where the disease is able to cross the species barrier and where both species are infected (Hadeler and Freedman [14]). However, the influence of harvesting practices on the dynamics of ecosystems affected by disease, with a few notable exceptions, has not been considered in this literature. It is well known that harvesting can put intense pressure on ecosystems and species and strongly influence their dynamics (Dong et al. [12]). Harvesting can not only reduce the prey or the predator population (Clark [9]), but can also stabilize (Goh et al. [13]), destabilize (Azar et al. [3], Cohn [10]), or induce fluctuations in predator-prey systems (Costa [11]). Moreover Brauer and Soudack [6], Xiao et al. [19] and Zhang et al. [21] show that harvesting can produce very complex dynamics such as chaotic and periodic solutions or limit cycles oscillations. Bairagi et al. [4] and Bhattacharyya and Mukhopadhyay [5] consider an epidemiological model with a disease in the prey and prey harvesting but as far as we know, the effect of harvesting a predator affected by a disease has never been studied in the literature. However, empirical observation of severe mortality in Pike (Esox lucius) due to the Viral Hemorrhagic Septicemia (VHS) epidemics in Canada and Continental Europe s waters shows that the case for a disease in an harvested predator is relevant. In particular, it is important in the formulation of harvesting policies to understand the response of a harvested predator-prey system to a disease in the predator. As shown by Bairagi et al. [4], harvesting by reducing the population of infected prey can be used to control the spread of a disease in the prey. They point out however that this strategy must be employed cautiously to avoid predator extinction. Indeed, a disease in the prey combined with prey harvesting can reduce drastically the food available to the predator and can lead the predator population to extinction. Auger et al. [2], even if they do not consider harvesting in their analysis, have also shown that a prey disease which would have evolved toward an endemic state in the absence of predators, can die out for a sufficiently high predation rate on the infected and hence more vulnerable prey. In this paper, we provide a mathematical framework to study the response of a harvested predator-prey system to a disease in the predator and proportional (constant effort) harvesting of predator. We combine a classical Lotka and Volterra predator-prey model with an eco-epidemiological model where predator disease is modeled by a Susceptible-Infected (SI) epidemic system. We show that harvest can be an effective strategy for controlling the spread of the disease. Whereas the disease can become endemic in the population at low harvest rates, it can always be completely eradicated by harvesting the predator at a sufficiently high rate. These results question the adoption of conservative harvest policies in order to enhance the resilience and stability of harvested ecological systems to emerging infectious diseases. 2

3 The rest of the paper is organized as follows. Section 2 introduces the eco-epidemiological model. Section 3 is devoted to the study of the possible equilibria. Section 4 discusses the role of harvesting as a disease control strategy and Section 5 concludes. 2 Formulation of the model 2.1 Basic predator-prey system Consider a simple predator prey system with constant effort harvesting of predator { Ṅt = rn t (1 N t /K) αn t P t P (1) t = αβn t P t µp t γep t where N t is the population of the prey and P t the population of predators at time t. In (1) we assume the prey grows according to a logistic growth with intrinsic growth rate r and carrying capacity K in the absence of predation. The predator consumes the prey at a rate proportional to the rate at which the predators and the prey meet with α a capture coefficient and β a coefficient of conversion of prey into predators. The predator mortality rate is µ. The predator is subject to constant effort harvesting where E denotes the constant harvesting effort and γ the catchability coefficient of the predators. We restrict our attention to biologically meaningful initial conditions for both species 0 < P (0) and 0 < N(0) < K The system (1) is globally asymptotically stable. Its globally asymptotically stable equilibrium is either a positive equilibrium state ˆQ = ( ˆN, ˆP ), where ˆN, ˆP > 0, that corresponds to a stable coexistence of both species, or a predator-free equilibrium Q 0 = (K, 0) that corresponds to the extinction of predator and the survival of prey at its carrying capacity. Let us introduce the basic reproduction number R 0 of the predator, which gives the expected number of predators produced by a single predator when introduced in a predatorfree environment. For the system (1), the basic reproduction number of the predator is defined as R 0 = αβk (2) µ + γe A single predator when introduced in a predator-free environment produces new predators at a rate αβk over its lifetime 1/(µ + γe). The basic reproduction number R 0 is a threshold parameter for the system (1) since if R 0 > 1 the predator can persist, whereas it cannot if R 0 < 1. In other words the maximal per capita growth rate of the predator population has to be positive for the persistence of the predator population. 3

4 P P = 0 P P = 0 Ṅ = 0 ˆP ˆQ Ṅ = 0 0 ˆN K N 0 K Q 0 N Figure 1: Equilibria of the predator-prey system More precisely, if R 0 > 1 then there exists a unique and globally stable positive equilibrium ˆQ = ( ˆN, ˆP ) where ˆN = K/R 0, ˆP = r α (1 1/K/R 0) and if R 0 1, then there is no positive equilibrium state and the predator-free equilibrium state Q 0 = (K, 0) is globally asymptotically stable. Notice that only if R 0 < 1, the predator is driven to extinction in finite time. It is worth to note that harvesting reduces the basic reproduction number of the predator 2 and thus its ability to persist. In any harvested predator-prey system there exists a critical value of harvesting effort given by E 0 = (αβk µ)/γ below which the predator will persist and above which it will die out. 2.2 The eco-epidemiological model Suppose now that the predator can be infected by a disease and divide the predators into two classes, susceptibles (non-infected) and infected, the population of each is given by S t and I t, respectively, so that the total population of predator at any time t is P t = S t + I t. We assume that the spread of the disease in the population of predators follows the classical law of mass action with λ as the transmission parameter. The susceptible and infected predators differ in their respective mortality rates µ and µ with an extra mortality due to the disease. Accordingly, we assume µ > µ. Susceptible and infected predators are subject to non-selective harvesting and a constant harvesting effort E applies indistinctly to both the susceptible and infected predators. However the susceptibles and infectives differ in their catchabilities. We assume that γ, the catchability coefficient of the infected predator, is greater than γ, the catchability coefficient of the susceptible predator. Indeed, for the same effort E the number of infected 4

5 individuals caught may be much higher than that of non-infected ones because infected individuals are less active than healthy ones (Bairagi et al. [4]). From the above assumptions, the dynamics of the ecosystem can now be described by the following eco-epidemiological model Ṅ t = rn t (1 Nt ) αn K t(s t + I t ) Ṡ t = αβn t S t µs t λi t S t γes t (3) I t = αβn t I t µ I t + λi t S t γ EI t We restrict our attention to biologically meaningful initial conditions 0 < S(0), 0 < I(0) and 0 < N(0) < K 3 Stability analysis and equilibria 3.1 Equilibria The equilibria of the epidemiological system (3) are obtained as solutions to the following equations rn(1 N ) = αn(p + I) K αβns µs γes = λis (4) αβni µ I γ EI = λis (5) (These equations correspond to the condition Ṅt = Ṡt = I t = 0). The system can have different (non trivial) equilibria: (i) a predator free-equilibrium Q 0 = (K, 0, 0), where the predator population dies out and the prey tends to its carrying capacity; (ii) a disease-free equilibrium (DFE) ˆQ = ( ˆN, Ŝ, 0), where ˆN, Ŝ > 0 and he disease eventually disappears from the predator population; (iii) an endemic positive equilibrium Q = (N, S, I ) where N, S, I > 0 and both infected and non infected predators coexist, and; (iv) an endemic semi-positive equilibrium ˆQ = ( ˆN, 0, Î ) where ˆN, I > 0 and the entire population of predators become infected by the disease. For further reference, define the basic reproduction number R 1 as the expected number of secondary cases produced, in a completely susceptible population, by a typical infected individual during its entire period of infectiousness. It describes the ability of an infectious disease to invade a population. For the system (3), R 1 is defined as the product of the infection rate and the expected lifetime of an infected inividual at a disease-free equilibrium ˆQ λŝ R 1 = µ + γ E αβ ˆN Over its entire lifetime 1/(µ + γ E αβ ˆN), a single infected predator introduced in a disease-free environment will produce secondary cases at a rate equal to λŝ. 5

6 It is a threshold parameter for the eco-epidemiological model. If R 1 < 1, then on average an infected predator produces less than one new infected over its expected lifetime, and the infection cannot grow. Conversely, if R 1 > 1, each infected predator produces, on average, more than one new infection, and the disease can invade the population. At the disease-free equilibrium, the disease dies out so that the total population of predators is ˆP = Ŝ. Therefore, the disease-free equilibrium ˆQ corresponds to the positive coexistence equilibrium of the classical predator-prey system (1) { Ṅt = rn t (1 N t /K) αn t P t P (6) t = αβn t P t µp t γep t and thus ˆN = K/R 0 and ˆP = r α (1 1/R 0) P P = 0 Ṅ = 0 0 Q 0 K ˆN N Figure 2: Predator-free equilibrium A necessary existence condition for this equilibrium is R 0 > 1. The basic reproduction number of the non infected predator has to be greater than one for the persistence of the non infected predator population. The basic reproduction number R 1 can be expressed in terms of R 0. We have R 1 = λ r (1 1/R α 0) ( ) αβk 1 R 0 At the semi-positive endemic equilibrium state ˆQ = ( ˆN, 0, Î ), the predator population becomes entirely infected by the disease. Therefore ˆP = Î and the equilibrium ˆQ correspond to the positive coexistence equilibrium of the following predator-prey system { Ṅt = rn t (1 N t /K) αn t P t P t = αβn t P t µ P t γ (7) EP t 6 1 R 0

7 P Ṅ = 0 Ṡ = 0 P ˆP ˆQ 0 N ˆN K N Figure 3: Disease-free equilibrium When compared to the classical predator-prey system (1), this system differs only in the terms µ P and γ EP in the second equation. Thus it admits a coexistence positive equilibrium given by: ˆN = K/R 0 and ˆP = r α (1 1/R 0) where R 0 denote the basic reproduction number for an infected predator R 0 = αβk µ + γ E P ˆP ˆP Ṅ = 0 I = 0 ˆQ P 0 ˆN N ˆN K N Figure 4: Endemic semi-positive equilibrium A necessary existence condition for this equilibrium is R 0 > 1. Because the basic reproduction number of the predators is always less for an infected predator than for a susceptible one, R 0 > 1 is a necessary condition for the persistence of the predator population. 7

8 At the positive endemic equilibrium Q = (N, S, I ) the following equalities hold:where P = S + I. From equation (4) and (5), we obtain that Therefore, and from equation (??), I = αβn µ γe λ S = αβn µ γ E λ P = (µ µ) λ + (γ γ) E λ N = K(1 α r P ) (8) (9) At the positive endemic equilibrium the total population of predators P increases whereas the population I of infectives decreases with harvesting effort. Accordingly, the proportion I /P of infected predators at the positive endemic equilibrium decreases with harvesting effort. This suggests that harvesting can be used as a strategy to control the spread of the disease and to enhance the resilience of the ecosystem. We provide a much more complete treatement to this question in Section 4. Using the basic reproduction numbers we can rewrite these expressions as: P = αβk [ 1 1 ] (10) λ R 0 R 0 = r α (1 1/R 0)/R 1 and ( N = K 1 α [ αβk 1 1 ]) r λ R 0 R 0 = K (1 (1 1/R 0 ) /R 1 ) From (8), we see that for the disease to become endemic we must have the following condition to be satisfied N > ˆN This condition is equivalent to the condition R 1 > 1. We can also observe that the population of predators is always less at an endemic equilibrium (that is at ˆQ or Q ) than at the disease-free equilibrium Q 0. Indeed for any constant harvesting effort E, ˆP = r (1 1/R α 0) is larger than P = r (1 1/R α 0)/R 1 and ˆP = r (1 α 1/R 0) if R 1 > 1 and R 0 > R 0 respectively. This conclusion is opposite to the conclusion obtained by Auger et al. [2] in their model with standard mass action law. Indeed, they obtain in the absence of harvesting that when a disease affects the predator, the population of predators at the disease-free equilibrium is always less than the predator population at the endemic equilibrium. 8

9 P Ṅ = 0 ˆP P Q 0 ˆN N K N Figure 5: Endemic positive equilibrium where Moreover from (9),a positive equilibrium population of susceptibles requires N < ˆN This condition is equivalent to the condition R 2 = R 2 < 1 λî αβ ˆN µ γe The number R 2 can be interpreted as the expected number of new infected cases produced, in a completely infected population, by a typical susceptible individual during its entire life span. If R 2 < 1, then on average a susceptible individual produces less than one new infected over its expected life span, and susceptible individual can then persist in the predator population. Conversely, if R 2 > 1, a susceptible predator, when introduced in a completely infected population produces, on average, more than one new infected individual, and susceptible individuals cannot persist in the population. The next proposition states the existence conditions for the possible equilibria. Proposition 1 The system (3) has a unique equilibrium. It is either (i) the predator-free equilibrium Q 0 if R 0 1, or; (ii) the disease-free equilibrium ˆQ if R 1 1, or; (iii) the endemic positive equilibrium Q if R 1 > 1 and R 2 < 1, or; (iv) the endemic semi-positive equilibrium ˆQ if R

10 Proof. First observe that R 1 = λ r (1 1/R α 0) ( ) and R 2 = αβk 1 R 0 1 R 0 λ r (1 α ( 1/R 0) ) αβk 1 R 0 1 R 0 Because R 0 > R 0, R 1 > 0 implies R 0 > 1. Similarly, R 2 > 0 implies R 0 > 1, and again because R 0 > R 0, this implies R 0 > 1. Moreover R 2 > R 1. Now, to prove that one and only one of these possible equilibria is the equilibrium of the system suppose that equilibrium (iii) does not exist. Then R 1 1 or R 2 1 (or both) or R 0 1. If R 0 1 then the system has (i) a unique equilibrium. If R 2 1 it is clear then R 1 > 1. Thus either R 1 1 or R 2 1. If R 2 1 then the system has (iv) as unique equilibrium. If R 1 1, then the system has (ii) as unique equilibrium. 3.2 Stability analysis In this section we investigate the local stability of system (3) around each of the possible equilibria. If the system has a different equilibrium from the positive endemic equilibrium as unique equilibrium, then the system is locally asymptotically stable around this equilibrium. Indeed, in these cases stability analysis reduces to investigating local stability a basic predator-prey system and local stability of these quilibria follows from standard arguments. It is thus sufficient to study the local stability of the system arond the positive endemic equilibrium to complete the local stability analysis of the eco-epidemiological model. Proposition 2 The system (3) is locally asymptotically stable around (i) the predator-free equilibrium Q 0 if R 0 1; (ii) the disease-free equilibrium ˆQ if R 1 1; (iii) the endemic positive equilibrium Q if R 1 > 1 and R 2 < 1; (iv) the endemic semi-positive equilibrium ˆQ if R 2 1. Proof. The variational matrix (3) of the system at the positive endemic positive equilibrium Q = (N, S, I ), where N, S, I > 0 is rn /K αn αn αβs 0 λs (11) αβi λi 0 The characteristic equation of the variational matrix (11) is given by ξ 3 + Aξ 2 + Bξ + C = 0 (12) 10

11 where A = rn /K B = λ 2 S I + α 2 βn (S + I ) C = rλ 2 N S I /K According to the Routh-Hurwitz stability criterion, all eigenvalues of the characteristic equation (12) have negative real part if A > 0, C > 0 and AB C > 0. It is easily checked that A and C are always positive. Moreover AB C = rα2 β K N 2 (S + I ) which is also always positive. Thus the eigenvalues of the characteristic equation all have negative real parts and the disease endemic positive equilibrium Q is locally asymptotically stable. Hence the proof. 4 Harvesting as a disease control strategy Suppose that system (1) has a positive coexistence equilibrium in the absence of harvesting. Then a disease free equilibrium can always be obtained by some harvesting strategy. The function R 0 = αβk is continuous decreasing in harvesting effort E such that µ+γe R 0 = αβk/µ in the absence of harvesting and such that R 0 tends to 0 as E tends to infinity. Assume K > αβ/µ so that, in the absence of harvesting, R 0 > 1. Then there exists some positive E 0 = (αβk µ)/γ such that R 0 > 1 for any E < E 0. In the absence of a disease, it is therefore possible to accomodate harvesting and the existence of stable equilibrium where both species coexist provided a reasonnable harvesting effort is applied to the predator. Now consider the function R 1 where ( ) λr 1 µ+γe α αβk R 1 = (µ µ) + (γ γ)e On domain [E, E 0 ], R 1 is a continuous and decreasing function of harvesting effort, that is harvesting enhance the resilience of the system to a infectious disease in the predator, and such that R 1 tends to 0 as E tends to E 0. It is therefore possible to find some reasonable harvesting strategy E < E 0 such that R 0 > 1 and R 1 1. In other words, it is always possible to find a reasonnable harvesting strategy that prevents the disease to become endemic in the predator population. A trivial case is where R 1 < 1 for any positive E < E 0. This case prevents the possibility for the disease to become endemic even in the absence of harvesting. The other case is where there exists a positive E 1 < E 0 such that R 1 > 1 for any E < E 1 and R 1 < 1 for any E > E 1. This case amounts to the case where R 1 > 1 in the absence of harvesting or, equivalently, that µ µ λ < r α (1 µ αβk ) 11

12 All other things being equal, the extramortality caused by the disease has to be not too high or its transmission rate has to be sufficiently high for the disease to become endemic in the predator population. In this case [ ] λ r 1 µ (µ µ) α αβk E 1 = λ rγ + αβk (γ γ) and for any harvesting strategy E 1 < E < E 0 such that R 0 > 1 and R 1 > 1. This strategy achieves a disease free equilibrium. Also notice that for harvesting strategies E < E 1 the system achieves an epidemic positive equilibrium in the long run. References [1] Anderson, R.M. and May, R., (1982). The invasion, persistence and spread of infectious diseases in animal and plant communities. Philosophical Transactions of the Royal Society B: Biological Sciences 314, [2] Auger, P., Mchich, R., Chowdhury, T., Sallet, G., Tchuente, M. and Chattopadhyay, J., (2009). Effects of a disease affecting a predator on the dynamics of a predator-prey system, Journal of Theoretical Biology 258, [3] Azar, C., Holmberg, J. and Lindgren, K., (1995). Stability analysis of harvesting in a predator-prey model. Journal of Theoretical Biology 174, [4] Bairagi, N., Chaudhury, S. and Chattopadhyay, J., (2009). Harvesting as a disease control measure in an eco-epidemiological system - a theoretical study, Mathematical Biosciences 217, [5] Bhattacharyya, R. and Mukhopadhyay, B., (2010) On an eco-epidemiological model with prey harvesting and predator switching: Local and global perspectives, Nonlinear Analysis: Real World Applications 11, [6] Brauer, F. and Soudack, A.C., (1981). Coexistence properties of some predator-prey systems under constant rate harvesting and stocking, Journal of Mathematical Biology 12, [7] Chattopadhyay, J. and Arino, O., (1999). A predator prey model with disease in the prey. Nonlinear Anal ysis 36, [8] Chattopadhyay, J. and Bairagi, N., (2001). Pelicans at risk in Salton sea - an ecoepidemiological study, Ecological Modelling, 136, [9] Clark, C.W., (1976). Mathematical bioeconomics: the optimal management of renewable resources (2nd ed. John Wiley and Sons: New York). 12

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