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Removal of infected prey prevent limit cycle oscillations in an infected prey–predator system—a mathematical study
Ecological Modelling 156 (2002) 113 /121 www.elsevier.com/locate/ecolmodel
Removal of infected prey prevent limit cycle oscillations in an infected prey predator system* a mathematical study /
/
J. Chattopadhyay a,*, R.R. Sarkar a, G. Ghosal b a
Embryology Research Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata 700 035, India b Sahapur Harendranath Vidyapith, 44, J.K. Pal Road, Kolkata 700 038, India
Received 14 November 2001; received in revised form 3 April 2002; accepted 16 April 2002
Abstract Predator /prey model with harvesting is well studied. The role of disease in such system has a great importance and can not be ignored. A mathematical model of a predator /prey system with infection on prey population is proposed and analyzed. In the formulation of the model, we assume that the predator response function is of Holling type II and the mode of disease transmission follows simple mass action law. Conditions for the persistence of the species is worked out. Our equilibrium and numerical analysis show that harvesting of infected prey population prevents the limit cycle oscillations and may be used as a biological control for the persistence of infected prey /predator dynamics. The applicability of the model in a real life situation is also discussed. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Predator /prey system; Infected prey; Force of infection; Harvest; Limit cycle; Controllability
1. Introduction A quantitative and qualitative understanding of the interaction of different species is crucial for the management of fisheries. Much research effort has been put into improving this understanding. Harvesting has generally a strong impact on the population dynamics of a harvested species. The severity of this impact depends on the nature of the implemented harvesting strategy which in turn may range from the rapid depletion to the complete preservation of a population. The study
* Corresponding author. Tel.: /91-33-5778086x3231; fax: / 91-33-5776680 E-mail address: [email protected] (J. Chattopadhyay).
of population dynamics with harvesting is a subject of mathematical bioeconomics, and it is related to the optimal management of renewable resources (Clark, 1990). The exploitation of biological resources and the harvest of population species are commonly practiced in fisheries, forestry, and wild life management. Problems related to the exploitation of multispecies systems are interesting and difficult both theoretically and practically. The problem of inter species competition between two species which obey the law of logistic growth has been considered by Gause (1935). But he did not study the effect of harvesting. Clark has also considered harvesting of a single species in a two fish ecologically competing population model. Modifying Clark’s model Chaudhuri (1986, 1988) studied combined harvest-
0304-3800/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 0 0 ( 0 2 ) 0 0 1 3 3 - 3
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ing and considered the perspectives of bioeconomics and dynamic optimization of a two species fishery. Brauer and Soudack (1981), Dai and Tang (1998) extended their study of constant rate of harvesting in a predator /prey systems to allow simultaneous harvesting of both species. They showed how to approximate the region of asymptotic stability in biological terms in the initial states that ultimately lead to coexistence of the two species and by efficient computer simulation they also performed the global dynamics of the system. The optimal harvesting policy in stochastic environment is also well studied branch in ecology. For example, the optimal harvesting rule for a lichen/ reindeer system with stochastic environment studied by Virtala (1996) and McClanahan (1995) studied the harvesting dynamics in an uncertain world by considering a simple food web model. Disease in predator /prey dynamics with harvesting is also an important issue. As far our knowledge goes no such study has been performed. The importance of transmissible disease in an ecological situation is not to be ignored. Little attention has been paid so far in this direction. To the best of our knowledge except for the papers of Hadeler and Freedman (1989), Freedman (1990), Beltrami and Carroll (1994), Venturino (1995), Xiao and Chen (2001), Chattopadhyay and Arino (1999), Chattopadhyay et al. (1999), Chattopadhyay and Pal (2002), no work has been carried out in such eco-epidemiological system. Suttle et al. (1990) showed by using electron microscopy that the viral disease can infect bacteria and even phytoplankton in coastal water. The fact is that parasites may modify the behavior of the infected member of the prey population. Holmes and Bethel (1972) indicated that this might happen in the following manner: reducing stamina, increasing constricuousness, disorientation and altering responses. The evidence of such a situation has been observed by many researchers. Lafferty and Morris (1996) observed that Kill fish (Fundulus parvipinnes) tends to come closer of the sea on contracting a disease in this way, which makes them more vulnerable to predation by birds. Arme and Owen (1967) and Williams (1967) observed the same behavior when sticklebacks (Gasterosteus
awleatus L.) infected by plerocarcoids (Schistocephalus solidus ). The effect of the disease in harvested predator / prey dynamics is important from economical viewpoint. The present paper deals with the problem of a harvested predator /prey system with an introduction of infection in prey population. The basic aim of this paper is to observe the dynamics of the system in such a situation and find out the conditions under which limit cycle oscillations can be prevented.
2. The mathematical model Let x(t) and z (t ) be the numbers of the susceptible and infected prey population per liter at time t. Let y (t ) be the number of the predator population per liter at time t. Let a (/0) be the intrinsic growth rate of the susceptible prey, K is the carrying capacity of the susceptible prey in the absence of predator and harvesting in the environment. Let c (/0) be the conversion factor denoting the number of newly born predators for each captured susceptible prey. g (/0) be the death rate of the predator. E1, E2 and E3 ( ]/0) be the harvesting efforts for the susceptible prey, predator and infected prey, respectively. Let l (/0) be the rate at which the prey population is infected and d (/0) is the rate at which the predator population is consuming the infected prey population. Let m ( /0) be the natural death rate of infected prey population other than predation. From the above assumptions to describe the dynamics of the system, the following set of differential equations can be written as: dx x bxy ax 1 E1 xlxz (1a) dt K 1 ax dy cbxy gy E2 ydyz (1b) dt 1 ax dz lxzdyzmzE3 z: (1c) dt Here the term bx /1/ax is the Holling type II functional response of the predator. The terms E1x , E2y and E3z represent the catch of the
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respective species. Actually it is more appropriate to take these terms as q1E1x , q2E2y and q3 E3z, where q1, q2 and q3 represent the catchability coefficients of the susceptible prey and predator, respectively. But, for the sake of mathematical simplicity, we have taken these terms to be unity. From Eq. (1a), we observe that the dynamics of the susceptible prey is governed by the logistic equation in the absence of predator and harvesting. System (1) has to be analyzed with the following initial conditions x(0)0;
y(0) 0;
z(0)]0:
(2)
Kuang and Freedman (1988) studied the dynamics of system (Eqs. (1a) and (1b)) with E1 and E2 / 0 and obtained conditions for which the system possesses a unique limit cycle oscillation in the first quadrant. Recently Srinivasu et al. (2001) studied the system (Eqs. (1a) and (1b)) with independent harvesting on both the species and showed that the system enters into a stable position from the limit cycle oscillation. They arrived at the conclusion that harvesting efforts may be used as a biological control for persistence of the species.
3. Some basic results We first observe that the right hand side of the system (1) is a smooth function of the variables (x , y , z ) and the parameters, as long as these quantities are non-negative, so local existence and uniqueness properties hold in the positive octant. From the first equation of system (1), it follows that x /0 is an invariant subset, that is x/0 if and only if x (t) /0 for some t . Thus x (t)/0 for all t if x (0) /0. Similar arguments follows for y /0 and z /0 from the second and third equations of system (1). Now we consider the boundedness of solutions of system (1). Theorem 3.1. If c B/1, all the solutions of system (1) which initiate in R3 are uniformly bounded . Proof.
Let us define a function
W xyz
115
(3)
The time derivative of Eq. (3) along the solutions of (1) is dW x (1 c)bxy ax 1 dt K 1 ax E1 x(E2 g)y(E3 m)z x E1 (E2 g)y(E3 m)z 5x a 1 K (provided cB 1):
(4)
For each D (/0), the following inequality holds dW DW dt x 5 x a 1 E1 D (E2 gD)y K (E3 mD)z
(5)
Now if we take D B/min{E2/g, E3/m} and the maximum value of the expression x [a (1/(x / K ))/E1/D ] is (a/E1/D )2K /2a then the Eq. (5) reduces to dW (a E1 D)2 K DW 5 : dt 2a Thus, we can find a constant L (/0) (say), such that dW DW 5L: dt Applying the theorem of differential inequality (Birkhoff and Rota, 1982), we obtain 0BW (x; y; z) 5
L (1eDt )W (x(0); y(0); z(0))eDt ; D
and for t 0/, we have 0BW 5
L : D
Hence all the solutions of (1) that initiate in {R3\0} are confined in the region
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L B (x; y; z) R3 : W o ; D
This equilibrium exists if aK(a E1 ) a
for any o /0 and for t large enough. Hence the theorem. Next we find all the biologically feasible equilibria admitted by the system (1) and study the dynamics of the system around each equilibrium. The equilibria of system (1) are the intersection pointsofthesusceptiblepreyisoclineonwhichx? /0, predatorisoclineonwhichy ?/0andtheinfectedprey isocline on which z ?/0. Whenever we have the harvesting effort of the susceptible prey, E1 /a , then thesystemadmitsonlyoneequilibrium,givenbyS0 / (0, 0, 0), which is the trivial equilibrium. If we have a /E1, then there exists another equilibrium on the boundary of the first octant, S1 /(K (1/E1/a ), 0, 0). Looking for the planar equilibria we observe that there are two planar equilibria of the system (1). The x /y planar equilibrium S2 (x; ¯ y; ¯ 0) exists, where x=E2+g/cb/a (E2+g ) and y ¯ ac=KfK[cba(gE2 )]=[cba(gE2 )]2 g cE1 =cba(gE2 ); if cba(gE2 ) and (gE2 )[aaK(aE1 )]cbK(aE1 )B0: The x /z planar equilibrium S3 (x?; ¯ 0; z?) ¯ exists, where x ?/ E 3 / m /l and z ? / 1/l {a(1/ E3/m /lK )/E1}, if l /a(E3/m)/K (a/E1). Further for an interior equilibrium, obtain, S4 /(x *, y*, z *), where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x B B2 4AC =2A; y lx(E3 m)=d; z 1=d(1ax)[(E2 g)(1ax)cbx] and A aad; B lcbdK aadK adblKdaE1 dK aldK(gE2 ); C adK b(E3 m)E1 dK ldK(gE2 ):
we
K[b(1 c) a(E2 g)] B
Bl
d(a E1 ) b(E3 m) d(E2 g)
(6)
and E3 m l
Bx B
E2 g cb a(E2 g)
(7)
4. Eigen value analysis and persistence of the system Let us first consider the trivial equilibrium of the system (1), S0 /(0, 0, 0). The variational matrix of system (1) at S0 is 2 3 0 aE1 0 5: (gE2 ) 0 (8) V0 40 0 0 (E3 m) The eigenvalues of the variational matrix V0 are m1 /a/E1, m2 //(g/E2) (B/0) and m3 //(E3/ m) (B/0). Clearly this steady state is asymptotically stable if and only if a B/E1, in this case it attracts all the feasible solutions. When a /E1, the trivial equilibrium is unstable (saddle) and there exists a steady state S1 /(K (1/(E1/a )), 0, 0. Hence we see that whenever the harvesting effort of susceptible prey remains below of its intrinsic growth, susceptible population will persist. Existence conditions of S1 lead us the following result. Theorem 4.1. If a /E1 hold , then the system (1) has a non-negative equilibrium S1 /(K (1/(E1/a )), 0, 0). The variational matrix of system (1) at S1 is The eigenvalues of the variational matrix V1 are m1? //a/E1, m2? //(g/E2)/(cbK (a/E1)/a/ aK (a/E1)) and m3? /lK (1/(E1/a ))/m. Clearly m1? B/0 and if m2? B/0 that is if (g/E2) [a/ aK (a/E1)]/cbK (a/E1)/0 and m3? B/0 that is if
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2 6aE1 6 6 6 V1 6 60 6 6 4 0
3 bK(a E1 ) E1 lK 1 7 a aK(a E1 ) a 7 7 7 cbK(a E1 ) 7: (gE2 ) 0 7 a aK(a E1 ) 7 7 E1 5 m 0 lK 1 a
l B/(a (E3/m )/K (a/E1)), then the system (1) is asymptotically stable around S1. When (g/E2) [a/aK (a/E1)]/cbK (a/E1) B/0, S1 is unstable (saddle) and there exists a steady state S2 (x; ¯ y; ¯ 0); which is free from infected prey. Further if l /(a (E3/m )/K (a/E1)), then also the predator and disease free prey steady state is unstable (saddle) and there exists a feasible predator free steady state S3 (x?; ¯ 0; z?): ¯ Now we summarize the results in the following theorem. Theorem 4.2. Existence of S2 implies extinction of S1 . The variational matrix of system (1) at S2 is) 2 3 p q r V2 4s 0 t 5; (10) 0 0 u q/ where p //(ax /K )/(abxy /(1/ax)2), (bx /1/ax ), r//lx , s /cby /(1/ax )2, t/dy and u /lx/dy/m. The eigenvalues of the variational matrix V2 are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m??1 p p2 4sq=2; m??2 p p2 4sq=2; and m??3 u: Since p B/0, the roots m1ƒ and m2ƒ have negative real part. Now if u B/0 that is if lBaK(aE1 ) a=K[b(1c)a(E2 g)]; then system (1) is asymptotically stable around S2. When l aK(aE1 )a=K[b(1c)a(E2 g)]; then S2 is unstable (saddle) and there exists a feasible nonnegative equilibrium S4 /(x *, y*, z *). The stability of the system around this equilibrium ensures the persistence of the populations.
117
(9)
The variational matrix of system (1) at S3 is 2 3 p? q? r? V3 40 s? 0 5; (11) t? u? v? where p?(ax=K); ¯ q?(bx?=1a ¯ x?); ¯ r?lx?; ¯ s?(gE2 )(cbx?=1a ¯ x?)d ¯ z?; ¯ t?lz?; ¯ u?dz? ¯ and v? lx?m: ¯ The eigenvalues of the variational matrix V3 are m??? 1 ((p?v?) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (p?v?)2 (t?r?p?v?)=2); m??? 2 ((p?v?) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (p?v?)2 (t?r?p?v?)=2) and m???=s ?. ¯ ; then Since p ?B/0, and if v? /0 that is x?Bm=l; ??? m??? and m have negative real part. Hence, the 1 2 system will be asymptotically stable around S3 if m??? 3 B0; that is, if l /d (a/E1)/b (E3/m)/d (E2/ g). When l B/d (a/E1)/b (E3/m)/d(E2/g ), then the predator free state is unstable (saddle) and there exists a feasible non-negative equilibrium S4 /(x *, y *, z *). Now we are in a position to state the following theorem. Theorem 4.3. If the force of infection is bounded in the region aK(aE1 )a=K[b(1c)a(E2 g)]=BlB d(aE1 )b(E3 m)=d(E2 g)]; all the three species will persist .
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Fig. 3. Prevention of limit cycle oscillation by increasing harvesting effort of infected prey. Fig. 1. Stable situation for low infection and less harvesting effort of infected prey.
positive equilibrium S4 as l varies in ((aK(a E1 )a)=(K[b(1c)a(E2 g)]); (d(aE1 ) b(E3 m))=(d(E2 g))); we recall that the stability properties of S4 depend on the susceptible population x *, which we shall rename as E3 m E2 g ; (13) jx; j l cb a(E2 g) To study the local stability of the system around S4, we shall follow the approach of Liu (1994). The characteristic equation of the variational matrix V4 is m¯ 3 Q1 (j)m¯ 2 Q2 (j)mQ ¯ 3 (j)0; where the coefficients Qi (j ), i/1, 2, 3 are
Fig. 2. Limit cycle oscillation for high force of infection and less harvesting effort of infected prey.
The variational matrix of system (1) at S4 is 3 2 ax abxy bx lx7 6 (1 ax)2 1 ax 7 6 K 7 6 V4 6 cby 7: 6 0 dy 7 5 4(1 ax)2 lz dz 0 (12) To consider the local stability analysis of the
Q1 (j)
aj K
abjy (1 aj)2
;
Q2 (j)d2 yzl2 jz
cb2 j (1 aj)3
aj abjy lbdyz Q3 (j) d2 yz 2 K (1 aj) 1 aj
ldcbjyzj : (1 aj)2
(14)
Denote A(j)(aby=(1aj)2 )j: Applying 2 /b (bj=(1aj) ); y2 (l=d)(j(m=l)) and/ 1
J. Chattopadhyay et al. / Ecological Modelling 156 (2002) 113 /121
/
z(1=d)[(E2 g)cb1 (1aj)]:/ We get
Table 1 Model parameter values
al m b1 j A(j) d l and then Q1 (j) A(j)
a K
j;
m Q2 (j) l j [(E2 g)cb1 (1aj)] l
ab1 m d
lb1 d
(1c)
j
m l
[(E2 g)cb1 (1aj)]:
Parameters
Values
a K b a E1 g c E2
0.9 per day 100 Numbers per l 0.0005 l per day 0.002 l 0.2 per day 0.00075 per day 0.1 0.0004 per day
Routh /Hurwitzcriterion,S4islocallyasymptotically stablein(b2 ,j0).Furthermore,accordingtotheresults we have a simple Hopf-bifurcation toward periodic solutionsforincreasingj,beingD(2)(j )B/0in(j0,b2). Of course, S4 is unstable in (j0, b2). Thus, we get the following theorem.
l2 j[(E2 g)cb1 (1aj)]; d Q3 (j) l
119
(15)
Obviously Q3(j )/0 for all j /((m /l ), b2), where b2 /(E2/g /cb/a (E2/g)). As for Q1(j ) have two cases (i) Q1(j )/0 for all j /((m/l), b2), where b2 /(ld/aKmb )/a (lKb/ad )), and (ii) Q1(j )B/0 for all j /(b2, b2). Moreover let us look at D(2) (j)Q1 (j)Q2 (j)Q3 (j) l2 a A(j) j j[(E2 g)cb1 (1aj)] d K a cbb1 l2 b 1 A(j) j K 1 aj d m (1ajc) j l [(E2 g)cb1 (1aj)]
Obviously, D(2)(b2)B/0 and D(2)(b2 )/0. Since D (j ) is continuous on (b2 , b2), a value j0 /(b2 , b2) must exist at which D(2)(j0)/0. Furthermore, a simple but tedious algebraic calculation shows that (2)
dD(2) (j) jjj0 "0: dj Hence D(2)(j )/0 in (b2 , j0) and, according to
Theorem 4.4. If l /((aK (a/E1)/a )/(K [b (1/ c)/a (E2/g)]), (d (a/E1)/b (E3/m))/(d (E2/g)), a simple Hopf-bifurcation occurs at j0 /(b2 , b2) for increasing j, i.e. the positive equilibrium S4 is locally asymptotically stable in (b2 , j0) and unstable in (j0, b2).
5. Numerical results To study the dynamics of the system around the positive interior steady state, we have numerically simulated the system of equations (1) for a range of parameter values around it. Except l , d , m and E3, all the parameter values are chosen from Srinivasu et al. (2001) (see Table 1). Our model equations (1) are same as Srinivasu et al. (2001) except the infected prey population. They concluded that limit cycle oscillations can be prevented by choosing the appropriate values of harvesting efforts of E1 and E2. For our study we choose m /0.4 and d/0.001. Both the force of infection (l (litre/nos.-day-)) and harvesting effort of infected prey (E3) are important parameters in deciding the population density of the dynamical behavior of the system. Thus, they have been studied over a wide range. For
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E3 /0.0002 and l /(0.2, 0.85), system (1) with initial condition (x0, y0, z0)/(8, 1485, 2), settles down to a steady state solution, depicting a stable situation (Fig. 1). Limit cycle oscillation of the system occurs at l/0.85 and E3 /0.0002 (Figs. 2 and 3). This limit cycle oscillations can be prevented by increasing the harvesting effort of infected prey from E3 /0.0002 to E3 /2. It is interesting to note that system considered by Srinivasu et al. (2001) is stable around positive equilibrium for the values given in Table 1. The system remains stable for the introduction of l / (0.2, 0.85) and for l beyond the value 0.85 system around positive equilibrium enters into limit cycle oscillation. This oscillatory behavior could be controlled by removal of infected prey in a considerable amount.
6. Conclusion In this paper, we have proposed a mathematical model on predator /prey system with infection on prey population. We considered the predator response function is of Holling type II and the mode of disease transmission follows the simple law of mass action. Limit cycle oscillation in harvested predator /prey system are common in nature and control of such oscillation is utmost important from ecological as well as economical viewpoint. Our analysis shows that there exists a critical value of the force of infection below which the dynamics of the system around its interior steady state is stable and above which is unstable. The oscillatory behavior due to enhancement of force of infection can be prevented by increasing the harvesting effort on infected prey population. Thus, we may conclude that harvesting of infected prey may be used as a biological control for the persistence in an infected prey /predator system. Cropper and DiResta (1999) studied the effects of harvesting on Florida commercial sponge population. They concluded that unregulated harvesting might lead to a decline in the sponge population and affects the benthic community composition. Also there are certain reports about disease in the sponge population which effects the sponge to bleach from the base up until the whole
sponge is completely white and then it just crumbles apart and ultimately kill it. The infection on sponge population also affects their predators. Thus, the effect of disease in such a harvested prey /predator system is very much important in both ecological and economical point of view. Thus, a proper harvesting strategy is needed for the persistence of the species. We hope that our mathematical model and its result will be applicable in such systems.
Acknowledgements The authors are grateful to the anonymous referee for his valuable suggestions. Special thanks also rendered to Professor S.E. Jorgensen, Editorin-Chief, Ecological Modeling, for his valuable suggestions and pointing out some useful references in this context.
References Arme, C., Owen, R.W., 1967. Infections of the three-spined stickleback Gasterostens acnleatus L., with the plerocercoid larvae of Schiotocephalus solidus (Muller 1776) with special reference to pathological effects. Parasitology 57, 301 /314. Beltrami, E., Carroll, T.O., 1994. Modelling the role of viral disease in recurrent phytoplankton blooms. J. Math. Biol. 32, 857 /863. Birkhoff, G., Rota, G.C., 1982. Ordinary Differential Equations. Ginn Boston. Brauer, F., Soudack, A.C., 1981. Coexistence properties of some predator prey systems under constant rate harvesting and stocking. J. Math. Biol. 12, 101 /114. Chattopadhyay, J., Arino, O., 1999. A predator /prey model with disease in the prey. Nonlinear Anal. 36, 747 /766. Chattopadhyay, J., Pal, S., 2002. Viral infection on phytoplankton zooplankton system */a mathematical model. Ecol. Model., 151, 15 /28. Chattopadhyay, J., Ghosal, G., Chaudhuri, K.S., 1999. Nonselective harvesting of a prey /predator community with infected prey. Korean J. Comp. Appl. Maths. 6, 601 /616. Chaudhuri, K., 1986. A bioeconomic model of harvesting a multispecies fishery. Ecol. Model. 32, 267 /279. Chaudhuri, K., 1988. Dynamic optimization of combined harvesting of a two species fishery. Ecol. Model. 32, 267 / 279. Clark, C.W., 1990. Mathematical Bioeconomics: The Optimal Management of Renewable Resources. Wiley, New York, USA.
J. Chattopadhyay et al. / Ecological Modelling 156 (2002) 113 /121 Cropper, W.P., Jr, DiResta, D., 1999. Simulation of a Biscayne Bay, Florida commercial sponge population: effects of harvesting after Hurricane Andrew. Ecol. Model. 118, 1 / 15. Dai, G., Tang, M., 1998. Coexistence region and global dynamics of a harvested predator prey system. Siam. J. Appl. Math. 58, 193 /210. Freedman, H.I., 1990. A model of predator /prey dynamics as modified by the action of parasite. Math. Biosci. 99, 143 / 155. Gause, G.F., 1935. La The´orie Mathe´matique de la Lutte pour la Vie. Harmann, Paris. Hadeler, K.P., Freedman, H.I., 1989. Predator /prey population with parasite infection. J. Math. Biol. 27, 609 /631. Holmes, J.C., Bethel, W.M., 1972. Modification of intermediate host behaviour by parasites. In: Cunning, E.V., Wright, C.A. (Eds.), Behavioural Aspects of Parasites Transmissions, Supplement Number 1, Zool. J. Linnean Soc., 51, 123 /149. Kuang, Y., Freedman, H.I., 1988. Uniqueness of limit cycles in Gause-type models of predator /prey systems. Math. Biosci. 88, 67 /84. Lafferty, K.D., Morris, A.K., 1996. Altered behaviour of parasitized killfish increases susceptibility to predation by bird final hosts. Ecology 77, 1390 /1397.
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Liu, W.M., 1994. Criterion of Hopf bifurcation without using eigenvalues. J. Math. Anal. Appl. 182, 250 /255. McClanahan, T.R., 1995. Harvesting in an uncertain world: impact of resource competition on harvesting dynamics. Ecol. Model. 80, 21 /26. Srinivasu, P.D.N., Ismail, S., Naidu, C.R., 2001. Global dynamics and controllability of a harvested prey /predator system. J. Biol. Syst. 9, 67 /79. Suttle, C., Charm, A., Cottrell, M., 1990. Infection of phytoplankton by viruses and reduction of primary productivity. Nature 347, 467 /469. Venturino, E., 1995. Epidemics in predator prey models: disease in the prey. In: Arino, O., Axelrod, D., Kimmel, M., Langlais, M. (Eds.), Mathematical Population Dynamics: Analysis of Heterogeneity, Theory of Epidemics, vol. 1. S. Wuerz, Winnipeg, pp. 381 /393. Virtala, M., 1996. Harvesting a lichen /reindeer system in an uncertain environment. Ecol. Model. 89, 209 /224. Williams, H.M., 1967. Helminth diseases of fish. Helminth Abstr. 36, 261 /295. Xiao, Y., Chen, L., 2001. Modelling and analysis of a predator /prey model with disease in the prey. Math. Biosci. 171, 59 /82.
Removal of infected prey prevent limit cycle oscillations in an infected prey predator system* a mathematical study /
/
J. Chattopadhyay a,*, R.R. Sarkar a, G. Ghosal b a
Embryology Research Unit, Indian Statistical Institute, 203, B.T. Road, Kolkata 700 035, India b Sahapur Harendranath Vidyapith, 44, J.K. Pal Road, Kolkata 700 038, India
Received 14 November 2001; received in revised form 3 April 2002; accepted 16 April 2002
Abstract Predator /prey model with harvesting is well studied. The role of disease in such system has a great importance and can not be ignored. A mathematical model of a predator /prey system with infection on prey population is proposed and analyzed. In the formulation of the model, we assume that the predator response function is of Holling type II and the mode of disease transmission follows simple mass action law. Conditions for the persistence of the species is worked out. Our equilibrium and numerical analysis show that harvesting of infected prey population prevents the limit cycle oscillations and may be used as a biological control for the persistence of infected prey /predator dynamics. The applicability of the model in a real life situation is also discussed. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Predator /prey system; Infected prey; Force of infection; Harvest; Limit cycle; Controllability
1. Introduction A quantitative and qualitative understanding of the interaction of different species is crucial for the management of fisheries. Much research effort has been put into improving this understanding. Harvesting has generally a strong impact on the population dynamics of a harvested species. The severity of this impact depends on the nature of the implemented harvesting strategy which in turn may range from the rapid depletion to the complete preservation of a population. The study
* Corresponding author. Tel.: /91-33-5778086x3231; fax: / 91-33-5776680 E-mail address: [email protected] (J. Chattopadhyay).
of population dynamics with harvesting is a subject of mathematical bioeconomics, and it is related to the optimal management of renewable resources (Clark, 1990). The exploitation of biological resources and the harvest of population species are commonly practiced in fisheries, forestry, and wild life management. Problems related to the exploitation of multispecies systems are interesting and difficult both theoretically and practically. The problem of inter species competition between two species which obey the law of logistic growth has been considered by Gause (1935). But he did not study the effect of harvesting. Clark has also considered harvesting of a single species in a two fish ecologically competing population model. Modifying Clark’s model Chaudhuri (1986, 1988) studied combined harvest-
0304-3800/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 0 0 ( 0 2 ) 0 0 1 3 3 - 3
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ing and considered the perspectives of bioeconomics and dynamic optimization of a two species fishery. Brauer and Soudack (1981), Dai and Tang (1998) extended their study of constant rate of harvesting in a predator /prey systems to allow simultaneous harvesting of both species. They showed how to approximate the region of asymptotic stability in biological terms in the initial states that ultimately lead to coexistence of the two species and by efficient computer simulation they also performed the global dynamics of the system. The optimal harvesting policy in stochastic environment is also well studied branch in ecology. For example, the optimal harvesting rule for a lichen/ reindeer system with stochastic environment studied by Virtala (1996) and McClanahan (1995) studied the harvesting dynamics in an uncertain world by considering a simple food web model. Disease in predator /prey dynamics with harvesting is also an important issue. As far our knowledge goes no such study has been performed. The importance of transmissible disease in an ecological situation is not to be ignored. Little attention has been paid so far in this direction. To the best of our knowledge except for the papers of Hadeler and Freedman (1989), Freedman (1990), Beltrami and Carroll (1994), Venturino (1995), Xiao and Chen (2001), Chattopadhyay and Arino (1999), Chattopadhyay et al. (1999), Chattopadhyay and Pal (2002), no work has been carried out in such eco-epidemiological system. Suttle et al. (1990) showed by using electron microscopy that the viral disease can infect bacteria and even phytoplankton in coastal water. The fact is that parasites may modify the behavior of the infected member of the prey population. Holmes and Bethel (1972) indicated that this might happen in the following manner: reducing stamina, increasing constricuousness, disorientation and altering responses. The evidence of such a situation has been observed by many researchers. Lafferty and Morris (1996) observed that Kill fish (Fundulus parvipinnes) tends to come closer of the sea on contracting a disease in this way, which makes them more vulnerable to predation by birds. Arme and Owen (1967) and Williams (1967) observed the same behavior when sticklebacks (Gasterosteus
awleatus L.) infected by plerocarcoids (Schistocephalus solidus ). The effect of the disease in harvested predator / prey dynamics is important from economical viewpoint. The present paper deals with the problem of a harvested predator /prey system with an introduction of infection in prey population. The basic aim of this paper is to observe the dynamics of the system in such a situation and find out the conditions under which limit cycle oscillations can be prevented.
2. The mathematical model Let x(t) and z (t ) be the numbers of the susceptible and infected prey population per liter at time t. Let y (t ) be the number of the predator population per liter at time t. Let a (/0) be the intrinsic growth rate of the susceptible prey, K is the carrying capacity of the susceptible prey in the absence of predator and harvesting in the environment. Let c (/0) be the conversion factor denoting the number of newly born predators for each captured susceptible prey. g (/0) be the death rate of the predator. E1, E2 and E3 ( ]/0) be the harvesting efforts for the susceptible prey, predator and infected prey, respectively. Let l (/0) be the rate at which the prey population is infected and d (/0) is the rate at which the predator population is consuming the infected prey population. Let m ( /0) be the natural death rate of infected prey population other than predation. From the above assumptions to describe the dynamics of the system, the following set of differential equations can be written as: dx x bxy ax 1 E1 xlxz (1a) dt K 1 ax dy cbxy gy E2 ydyz (1b) dt 1 ax dz lxzdyzmzE3 z: (1c) dt Here the term bx /1/ax is the Holling type II functional response of the predator. The terms E1x , E2y and E3z represent the catch of the
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respective species. Actually it is more appropriate to take these terms as q1E1x , q2E2y and q3 E3z, where q1, q2 and q3 represent the catchability coefficients of the susceptible prey and predator, respectively. But, for the sake of mathematical simplicity, we have taken these terms to be unity. From Eq. (1a), we observe that the dynamics of the susceptible prey is governed by the logistic equation in the absence of predator and harvesting. System (1) has to be analyzed with the following initial conditions x(0)0;
y(0) 0;
z(0)]0:
(2)
Kuang and Freedman (1988) studied the dynamics of system (Eqs. (1a) and (1b)) with E1 and E2 / 0 and obtained conditions for which the system possesses a unique limit cycle oscillation in the first quadrant. Recently Srinivasu et al. (2001) studied the system (Eqs. (1a) and (1b)) with independent harvesting on both the species and showed that the system enters into a stable position from the limit cycle oscillation. They arrived at the conclusion that harvesting efforts may be used as a biological control for persistence of the species.
3. Some basic results We first observe that the right hand side of the system (1) is a smooth function of the variables (x , y , z ) and the parameters, as long as these quantities are non-negative, so local existence and uniqueness properties hold in the positive octant. From the first equation of system (1), it follows that x /0 is an invariant subset, that is x/0 if and only if x (t) /0 for some t . Thus x (t)/0 for all t if x (0) /0. Similar arguments follows for y /0 and z /0 from the second and third equations of system (1). Now we consider the boundedness of solutions of system (1). Theorem 3.1. If c B/1, all the solutions of system (1) which initiate in R3 are uniformly bounded . Proof.
Let us define a function
W xyz
115
(3)
The time derivative of Eq. (3) along the solutions of (1) is dW x (1 c)bxy ax 1 dt K 1 ax E1 x(E2 g)y(E3 m)z x E1 (E2 g)y(E3 m)z 5x a 1 K (provided cB 1):
(4)
For each D (/0), the following inequality holds dW DW dt x 5 x a 1 E1 D (E2 gD)y K (E3 mD)z
(5)
Now if we take D B/min{E2/g, E3/m} and the maximum value of the expression x [a (1/(x / K ))/E1/D ] is (a/E1/D )2K /2a then the Eq. (5) reduces to dW (a E1 D)2 K DW 5 : dt 2a Thus, we can find a constant L (/0) (say), such that dW DW 5L: dt Applying the theorem of differential inequality (Birkhoff and Rota, 1982), we obtain 0BW (x; y; z) 5
L (1eDt )W (x(0); y(0); z(0))eDt ; D
and for t 0/, we have 0BW 5
L : D
Hence all the solutions of (1) that initiate in {R3\0} are confined in the region
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L B (x; y; z) R3 : W o ; D
This equilibrium exists if aK(a E1 ) a
for any o /0 and for t large enough. Hence the theorem. Next we find all the biologically feasible equilibria admitted by the system (1) and study the dynamics of the system around each equilibrium. The equilibria of system (1) are the intersection pointsofthesusceptiblepreyisoclineonwhichx? /0, predatorisoclineonwhichy ?/0andtheinfectedprey isocline on which z ?/0. Whenever we have the harvesting effort of the susceptible prey, E1 /a , then thesystemadmitsonlyoneequilibrium,givenbyS0 / (0, 0, 0), which is the trivial equilibrium. If we have a /E1, then there exists another equilibrium on the boundary of the first octant, S1 /(K (1/E1/a ), 0, 0). Looking for the planar equilibria we observe that there are two planar equilibria of the system (1). The x /y planar equilibrium S2 (x; ¯ y; ¯ 0) exists, where x=E2+g/cb/a (E2+g ) and y ¯ ac=KfK[cba(gE2 )]=[cba(gE2 )]2 g cE1 =cba(gE2 ); if cba(gE2 ) and (gE2 )[aaK(aE1 )]cbK(aE1 )B0: The x /z planar equilibrium S3 (x?; ¯ 0; z?) ¯ exists, where x ?/ E 3 / m /l and z ? / 1/l {a(1/ E3/m /lK )/E1}, if l /a(E3/m)/K (a/E1). Further for an interior equilibrium, obtain, S4 /(x *, y*, z *), where pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x B B2 4AC =2A; y lx(E3 m)=d; z 1=d(1ax)[(E2 g)(1ax)cbx] and A aad; B lcbdK aadK adblKdaE1 dK aldK(gE2 ); C adK b(E3 m)E1 dK ldK(gE2 ):
we
K[b(1 c) a(E2 g)] B
Bl
d(a E1 ) b(E3 m) d(E2 g)
(6)
and E3 m l
Bx B
E2 g cb a(E2 g)
(7)
4. Eigen value analysis and persistence of the system Let us first consider the trivial equilibrium of the system (1), S0 /(0, 0, 0). The variational matrix of system (1) at S0 is 2 3 0 aE1 0 5: (gE2 ) 0 (8) V0 40 0 0 (E3 m) The eigenvalues of the variational matrix V0 are m1 /a/E1, m2 //(g/E2) (B/0) and m3 //(E3/ m) (B/0). Clearly this steady state is asymptotically stable if and only if a B/E1, in this case it attracts all the feasible solutions. When a /E1, the trivial equilibrium is unstable (saddle) and there exists a steady state S1 /(K (1/(E1/a )), 0, 0. Hence we see that whenever the harvesting effort of susceptible prey remains below of its intrinsic growth, susceptible population will persist. Existence conditions of S1 lead us the following result. Theorem 4.1. If a /E1 hold , then the system (1) has a non-negative equilibrium S1 /(K (1/(E1/a )), 0, 0). The variational matrix of system (1) at S1 is The eigenvalues of the variational matrix V1 are m1? //a/E1, m2? //(g/E2)/(cbK (a/E1)/a/ aK (a/E1)) and m3? /lK (1/(E1/a ))/m. Clearly m1? B/0 and if m2? B/0 that is if (g/E2) [a/ aK (a/E1)]/cbK (a/E1)/0 and m3? B/0 that is if
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2 6aE1 6 6 6 V1 6 60 6 6 4 0
3 bK(a E1 ) E1 lK 1 7 a aK(a E1 ) a 7 7 7 cbK(a E1 ) 7: (gE2 ) 0 7 a aK(a E1 ) 7 7 E1 5 m 0 lK 1 a
l B/(a (E3/m )/K (a/E1)), then the system (1) is asymptotically stable around S1. When (g/E2) [a/aK (a/E1)]/cbK (a/E1) B/0, S1 is unstable (saddle) and there exists a steady state S2 (x; ¯ y; ¯ 0); which is free from infected prey. Further if l /(a (E3/m )/K (a/E1)), then also the predator and disease free prey steady state is unstable (saddle) and there exists a feasible predator free steady state S3 (x?; ¯ 0; z?): ¯ Now we summarize the results in the following theorem. Theorem 4.2. Existence of S2 implies extinction of S1 . The variational matrix of system (1) at S2 is) 2 3 p q r V2 4s 0 t 5; (10) 0 0 u q/ where p //(ax /K )/(abxy /(1/ax)2), (bx /1/ax ), r//lx , s /cby /(1/ax )2, t/dy and u /lx/dy/m. The eigenvalues of the variational matrix V2 are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m??1 p p2 4sq=2; m??2 p p2 4sq=2; and m??3 u: Since p B/0, the roots m1ƒ and m2ƒ have negative real part. Now if u B/0 that is if lBaK(aE1 ) a=K[b(1c)a(E2 g)]; then system (1) is asymptotically stable around S2. When l aK(aE1 )a=K[b(1c)a(E2 g)]; then S2 is unstable (saddle) and there exists a feasible nonnegative equilibrium S4 /(x *, y*, z *). The stability of the system around this equilibrium ensures the persistence of the populations.
117
(9)
The variational matrix of system (1) at S3 is 2 3 p? q? r? V3 40 s? 0 5; (11) t? u? v? where p?(ax=K); ¯ q?(bx?=1a ¯ x?); ¯ r?lx?; ¯ s?(gE2 )(cbx?=1a ¯ x?)d ¯ z?; ¯ t?lz?; ¯ u?dz? ¯ and v? lx?m: ¯ The eigenvalues of the variational matrix V3 are m??? 1 ((p?v?) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (p?v?)2 (t?r?p?v?)=2); m??? 2 ((p?v?) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (p?v?)2 (t?r?p?v?)=2) and m???=s ?. ¯ ; then Since p ?B/0, and if v? /0 that is x?Bm=l; ??? m??? and m have negative real part. Hence, the 1 2 system will be asymptotically stable around S3 if m??? 3 B0; that is, if l /d (a/E1)/b (E3/m)/d (E2/ g). When l B/d (a/E1)/b (E3/m)/d(E2/g ), then the predator free state is unstable (saddle) and there exists a feasible non-negative equilibrium S4 /(x *, y *, z *). Now we are in a position to state the following theorem. Theorem 4.3. If the force of infection is bounded in the region aK(aE1 )a=K[b(1c)a(E2 g)]=BlB d(aE1 )b(E3 m)=d(E2 g)]; all the three species will persist .
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Fig. 3. Prevention of limit cycle oscillation by increasing harvesting effort of infected prey. Fig. 1. Stable situation for low infection and less harvesting effort of infected prey.
positive equilibrium S4 as l varies in ((aK(a E1 )a)=(K[b(1c)a(E2 g)]); (d(aE1 ) b(E3 m))=(d(E2 g))); we recall that the stability properties of S4 depend on the susceptible population x *, which we shall rename as E3 m E2 g ; (13) jx; j l cb a(E2 g) To study the local stability of the system around S4, we shall follow the approach of Liu (1994). The characteristic equation of the variational matrix V4 is m¯ 3 Q1 (j)m¯ 2 Q2 (j)mQ ¯ 3 (j)0; where the coefficients Qi (j ), i/1, 2, 3 are
Fig. 2. Limit cycle oscillation for high force of infection and less harvesting effort of infected prey.
The variational matrix of system (1) at S4 is 3 2 ax abxy bx lx7 6 (1 ax)2 1 ax 7 6 K 7 6 V4 6 cby 7: 6 0 dy 7 5 4(1 ax)2 lz dz 0 (12) To consider the local stability analysis of the
Q1 (j)
aj K
abjy (1 aj)2
;
Q2 (j)d2 yzl2 jz
cb2 j (1 aj)3
aj abjy lbdyz Q3 (j) d2 yz 2 K (1 aj) 1 aj
ldcbjyzj : (1 aj)2
(14)
Denote A(j)(aby=(1aj)2 )j: Applying 2 /b (bj=(1aj) ); y2 (l=d)(j(m=l)) and/ 1
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/
z(1=d)[(E2 g)cb1 (1aj)]:/ We get
Table 1 Model parameter values
al m b1 j A(j) d l and then Q1 (j) A(j)
a K
j;
m Q2 (j) l j [(E2 g)cb1 (1aj)] l
ab1 m d
lb1 d
(1c)
j
m l
[(E2 g)cb1 (1aj)]:
Parameters
Values
a K b a E1 g c E2
0.9 per day 100 Numbers per l 0.0005 l per day 0.002 l 0.2 per day 0.00075 per day 0.1 0.0004 per day
Routh /Hurwitzcriterion,S4islocallyasymptotically stablein(b2 ,j0).Furthermore,accordingtotheresults we have a simple Hopf-bifurcation toward periodic solutionsforincreasingj,beingD(2)(j )B/0in(j0,b2). Of course, S4 is unstable in (j0, b2). Thus, we get the following theorem.
l2 j[(E2 g)cb1 (1aj)]; d Q3 (j) l
119
(15)
Obviously Q3(j )/0 for all j /((m /l ), b2), where b2 /(E2/g /cb/a (E2/g)). As for Q1(j ) have two cases (i) Q1(j )/0 for all j /((m/l), b2), where b2 /(ld/aKmb )/a (lKb/ad )), and (ii) Q1(j )B/0 for all j /(b2, b2). Moreover let us look at D(2) (j)Q1 (j)Q2 (j)Q3 (j) l2 a A(j) j j[(E2 g)cb1 (1aj)] d K a cbb1 l2 b 1 A(j) j K 1 aj d m (1ajc) j l [(E2 g)cb1 (1aj)]
Obviously, D(2)(b2)B/0 and D(2)(b2 )/0. Since D (j ) is continuous on (b2 , b2), a value j0 /(b2 , b2) must exist at which D(2)(j0)/0. Furthermore, a simple but tedious algebraic calculation shows that (2)
dD(2) (j) jjj0 "0: dj Hence D(2)(j )/0 in (b2 , j0) and, according to
Theorem 4.4. If l /((aK (a/E1)/a )/(K [b (1/ c)/a (E2/g)]), (d (a/E1)/b (E3/m))/(d (E2/g)), a simple Hopf-bifurcation occurs at j0 /(b2 , b2) for increasing j, i.e. the positive equilibrium S4 is locally asymptotically stable in (b2 , j0) and unstable in (j0, b2).
5. Numerical results To study the dynamics of the system around the positive interior steady state, we have numerically simulated the system of equations (1) for a range of parameter values around it. Except l , d , m and E3, all the parameter values are chosen from Srinivasu et al. (2001) (see Table 1). Our model equations (1) are same as Srinivasu et al. (2001) except the infected prey population. They concluded that limit cycle oscillations can be prevented by choosing the appropriate values of harvesting efforts of E1 and E2. For our study we choose m /0.4 and d/0.001. Both the force of infection (l (litre/nos.-day-)) and harvesting effort of infected prey (E3) are important parameters in deciding the population density of the dynamical behavior of the system. Thus, they have been studied over a wide range. For
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E3 /0.0002 and l /(0.2, 0.85), system (1) with initial condition (x0, y0, z0)/(8, 1485, 2), settles down to a steady state solution, depicting a stable situation (Fig. 1). Limit cycle oscillation of the system occurs at l/0.85 and E3 /0.0002 (Figs. 2 and 3). This limit cycle oscillations can be prevented by increasing the harvesting effort of infected prey from E3 /0.0002 to E3 /2. It is interesting to note that system considered by Srinivasu et al. (2001) is stable around positive equilibrium for the values given in Table 1. The system remains stable for the introduction of l / (0.2, 0.85) and for l beyond the value 0.85 system around positive equilibrium enters into limit cycle oscillation. This oscillatory behavior could be controlled by removal of infected prey in a considerable amount.
6. Conclusion In this paper, we have proposed a mathematical model on predator /prey system with infection on prey population. We considered the predator response function is of Holling type II and the mode of disease transmission follows the simple law of mass action. Limit cycle oscillation in harvested predator /prey system are common in nature and control of such oscillation is utmost important from ecological as well as economical viewpoint. Our analysis shows that there exists a critical value of the force of infection below which the dynamics of the system around its interior steady state is stable and above which is unstable. The oscillatory behavior due to enhancement of force of infection can be prevented by increasing the harvesting effort on infected prey population. Thus, we may conclude that harvesting of infected prey may be used as a biological control for the persistence in an infected prey /predator system. Cropper and DiResta (1999) studied the effects of harvesting on Florida commercial sponge population. They concluded that unregulated harvesting might lead to a decline in the sponge population and affects the benthic community composition. Also there are certain reports about disease in the sponge population which effects the sponge to bleach from the base up until the whole
sponge is completely white and then it just crumbles apart and ultimately kill it. The infection on sponge population also affects their predators. Thus, the effect of disease in such a harvested prey /predator system is very much important in both ecological and economical point of view. Thus, a proper harvesting strategy is needed for the persistence of the species. We hope that our mathematical model and its result will be applicable in such systems.
Acknowledgements The authors are grateful to the anonymous referee for his valuable suggestions. Special thanks also rendered to Professor S.E. Jorgensen, Editorin-Chief, Ecological Modeling, for his valuable suggestions and pointing out some useful references in this context.
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