BIFURCATION ANALYSIS OF A MODEL OF THREE-LEVEL FOOD CHAIN IN A MANGROVE ECOSYSTEM

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1 U.P.B. Sci. Bull., Series A, Vol. 78, Iss. 4, 2016 ISSN BIFURCATION ANALYSIS OF A MODEL OF THREE-LEVEL FOOD CHAIN IN A MANGROVE ECOSYSTEM Cristina Bercia 1 and Romeo Bercia 2 In this paper, we consider a three-dimensional non-linear dynamical system of predator-prey type with 12 parameters which models a food-chain in a mangrove ecosystem. Our goal is to study the dynamical properties of the model with constant rate harvesting on the top-predator. It is shown that transcritical, saddle-node and supercritical Hopf ifurcations occur when one parameter is varied. By numerical integration of the system, we plot the phase portraits for the important types of dynamics and we show the presence of a stale limit cycle. We deduce the controlling role of the harvesting rate upon the stale equlirium or periodic state of coexistence of the species. Keywords: Predator-prey system, Staility, Limit cycle, Bifurcation. 1. Introduction We consider a three dimensional ordinary differential autonomous system which models a food chain in a mangrove ecosystem with detritus recycling and with constant rate harvesting of top predators. The three levels of the food chain are the detritus (x(t) denotes its density at time t) which consists of algal species and leaves of the mangrove plants, then detritivores (y(t))- unicellular animals, crustacean, amphiopod and others (see [4]) which depend on rich detritus-ase and the predators of detritivores (z(t))-fish and prawn. Some of the predators have commercial value and undergo harvesting. The system has the following form: x (t) = x 0 ax βxye x + γz y (t) = β 1 xye x d 1 y cyz k + y z (t) = c 1yz k + y d 2z hz with the initial conditions x(0), y(0), z(0) 0. (1) 1 Lecturer, Faculty of Applied Sciences, University Politehnica of Bucharest, Romania, cristina ercia@yahoo.co.uk 2 Lecturer, Faculty of Applied Sciences, University Politehnica of Bucharest, Romania 105

2 106 Cristina Bercia, Romeo Bercia The main assumptions of the model are the following: 1. The detritus has a constant input rate (x 0 ) maintained y the decomposed mangrove leaves. [4] Also the detritus is washout from the system with a certain rate (a) and is supplied with dead organic matter converted into detritus y the action of the micro-organisms. 2. The interaction etween detritus and detritivores is different from the classical models of predator-prey in that the detritivores response function is not a monotone incresing function of prey density, ut rather is only monotone increasing until some critical density and then ecomes monotone decreasing. (as it is descried in [3] etween phytoplankton and zooplankton). This phenomenon is called group defense in which predation (in our case y detritivores) decreases when the density of the prey population is sufficiently large, which is also related to the nutrient uptake inhiition phenomenon in chemical kinetics (see [1]). One of the nonmonotone functional response introduced (see [6]) is p(x) = βxe x where 1 > 0 is the density of the prey at which predation reaches its maximum. 3. The amount of detritivores consumed y the top predator is assumed to follow a Holling type-ii functional response. It exhiits saturation effect when y-population is aundent and k is the half-saturation constant. The top predators (z(t)) are harvested with a constant rate, h. 4. The detritus-detritivores conversion rate is less then the detritus uptake rate, so β 1 < β. 5. For the same reason, the detritivore-predator conversion rate is c 1 < c. 6. The detritus recycle rate due to the death of predators, γ is less than d 2 + h. First we study the oundedness of the solutions of the system (1). Proposition 1.1. The first octant R 3 + is positively invariant under the flow generated y the system (1). Proof. Let e (v 1, v 2, v 3 ) the vector field which defines the differential system (1). We study the vector-field on the oundaries of the first octant. In Oxzplane, v 2 (x, 0, z) = 0, therefore all trajectories which initiate in this plane, remain in it, for any t 0, so the plane y = 0 is an invariant set for the system. In the same situation is the plane z = 0 ecause v 3 (x, y, 0) = 0. But trajectories which start in Oyz-plane, y, z > 0 are directed towards the interior of the first octant, since v 1 (0, y, z) = x 0 + γz > 0. In consequence all solutions with x(0), y(0), z(0) 0, remain in the first octant. Proposition 1.2. With the hypothesis upon the parameters 4,5 and 6, every solution of the system (1) with positive initial values, is ounded. Proof. Let (x(t), y(t), z(t)) e a solution of the system with positive initial conditions. We deduce that (x+y+z) (t) < x 0 ax(t)+(γ d 2 h)z(t) d 1 y(t), t 0. If M = min(a, d 2 + h γ, d 1 ), then (x + y + z) < x 0 M(x + y + z). Next we denote u(t) = (x + y + z)(t). By multiplying the last inequality with

3 Bifurcation Analysis of a Model of Three-level Food Chain in a Mangrove Ecosystem 107 e Mt, we get that u verifies (ue Mt ) x 0 e Mt < 0, t 0, or, equivalently from integration, u(t) (u(0) x 0 M )e Mt + x 0, that is u(t) max(u(0), x 0 ), t 0. M M Therefore, any solution of the system (1) starting in R 3 +, is ounded. Let us nondimensionalize the system (1) with the following scaling: t at; y β cβ y; z z and the system ecomes a a 2 x (t) = x xye x + µz (2) y (t) = xye x y yz K + y z (t) = ryz K + y Hz where = x 0 a ; = β 1 ; D a 1 = d 1 a ; K = kβ ; r = c 1 a a ; H = d 2+h ; µ = γa are positive a cβ constants. For simplicity, we keep the ecological implications of parameters, µ,,, K, r and H the same as x 0, γ, β 1, d 1, k, c 1 and h, respectively. In consequence, in the following study, we assume that µ < H, < 1 and r < Staility and ifurcation analysis Now we determine the location and the existence criteria for the equiliria of the system (2) in the first octant. Proposition 2.1. a) There is an axial equlirium E 0 (, 0, 0), for any values of the parameters; ) If e < < e and > 1, the system has two oundary equiliria E i (x i, yi, 0), i = 1, 2, where x 1 < x 2 are the two solutions of the equation xe x = (3) and yi = ( x i ),i = 1, 2; c) If > e, there exists only one oundary equilirium E 1 (x 1, y1, 0) with x 1 < 1 and y 1 given in ); d) If < e, there are no oundary equiliria in R 3 +. Proof. ) Un equilirium with z = 0 has its first component, solution of the equation xe x =. A simple classical analysis shows that the equation has solutions iff e and in the case with strict inequality there are two solutions x 1 < x 2 which are smaller than when e <, > 1. c) The condition > e is equivalent with x 1 < < x 2 and in consequence, only E 1 is in the first octant. We take the case when ( )M such that xe x x=m >, which, from the second equation in system (2), means that otherwise y could not survive on the prey at any density in the asence of z. In the following study, e and this is the case when the equation (3) posses at least one solution.

4 108 Cristina Bercia, Romeo Bercia Next we give a sufficient existence criterion for the interior equilirium E 3 (x 3, y 3, z 3 ). A simple calculation shows that x 3 is a solution of the equation: g(x) := K H µr r H xe x + x + µrk r H = 0, (4) y 3 = HK r H and z 3 = (x 3 e x 3 ) rk r H. Remark 2.1. We assume that r > H from now on, that is the detritivorepredator conversion rate has to e greater than the harvesting rate, so that there exists un interior equilirium which is the case with iological relevance. Proposition 2.2. i) If < e and > 1 (i.e. the system has two oundary equiliria) and H 2 < H < H 1, then it is at least one interior equilirium, E 3, where H i := r( x i ) i = 1, 2; (5) K + ( x i ), ii) If > e (i.e. there exists only one oundary equilirium) and H < H 1, then it is also at least one interior equilirium, E 3. iii) When the system has two oundary equiliria, H 2 < H 1. Proof. First of all note that z 3 > 0 is equivalent to x 3 e x 3 > or x 3 (x 1, x 2), where x 1 < x 2 are the two solutions of the equation xe x =. Then notice that in equation (4) the coefficient H µr > 0, due to the assumptions (4,5,6) of the model. A sufficient condition for equation (4) to have a solution x 3 (x 1, x 2) is g(x 1)g(x 2) < 0 or in the form x 1 < KH (r H) < x 2 which in case i) ecomes H 2 < H < H 1 and in case ii), H < H 1. We shall see that for H = H 1, E 3 = E 1 and in case i), for H = H 2, E 3 = E 2. Corollary 2.1. If, in addition to the conditions in Proposition 2.2 we require Kµ + e2 H < r =: H K + e 2 0, (6) it follows that the equation g(x) = 0 has only one solution x 3 (x 1, x 2), that is the equilirium E 3 exists and it is unique. Otherwise, g(x) = 0 may have maximum three positive solutions, so there are maximum three interior equiliria. Proof. g (x) = 1 + K H µr (1 r H x)e x attains its minimum m = 1 K(H µr) e 2 (r H) at x = 2. Because g (0) > 0 and lim g (x) > 0, it turns out that if m > 0 x (the condition (6)), then g is increasing. If m < 0, then g (x) = 0 has two solutions and in consequence, the equation g(x) = 0 may have three positive solutions.

5 Bifurcation Analysis of a Model of Three-level Food Chain in a Mangrove Ecosystem 109 Next we discuss the dynamics of the system (2) in the neighorhood of each equilirium. The Jacoian matrix of the linearization of the system is Dv D(x, y, z) = Evaluating 1 ye x (1 x) xe x µ ye x (1 x) xe x Kz (y+k) 2 Dv D(x,y,z) Proposition 2.3. i) If 0 rkz (y+k) 2 y y+k ry y+k H at each equilirium, we get the following results: < e (7), the equilirium E 0 is a hyperolic stale node. If > e, E 0 is a hyperolic saddle. In any case, it is attractive in two directions in the plane y = 0; ii) The oundary equilirium E 1 is locally asymptotically stale for H > H 1 and it is a hyperolic saddle for H < H 1 ; iii) E 2 is a hyperolic saddle for the values of the parameters which ensure its existence, namely < e, > 1 and for any H. Proof. i) The eigenvalues of the Jacoian matrix evaluated at E 0 are λ 1 = 1, λ 2 = H, λ 3 = e and the eigenvectors corresponding to λ 1,2 < 0 are u 1 = (1, 0, 0), u 2 = (µ, 0, 1 H) which imply i). ii),iii) For E i, i = 1, 2, the eigenvalues are λ i 1 = r( x i ) H and K +( x i ) λi 2,3 are the roots of the equation λ 2 + (1 + p i )λ + p i = 0 with p i = ( x i )(1 x i ). x i Hence, p 1 > 0 and in consequence, Re(λ 1 2,3) < 0, ut p 2 < 0 so, λ 2 2λ 2 3 < 0 and E 2 ecomes a saddle. The eigenvalues of the Jacoian matrix at the interior equilirium E 3 (x 3, y 3, z 3 ) verify the characteristic equation where A 1 = H r λ 3 + A 1 λ 2 + A 2 λ + A 3 = 0 (8) H r(r H) e x 3 [Kr x 3 ((r H) + Kr)]; A 2 = H r (x 3e x 3 )[r H 1 KH r H e x 3 (1 x 3 )] + KH r H e 2x 3 x 3 (1 x 3 ); (9) A 3 = (x 3 e x 3 )(r H)[ KH r H e x 3 (1 x 3 )( H r µ) + H r ]. The necessary and sufficient condition for E 3 to e asymptotically stale is given y the Routh-Hurwitz criterion, i.e. A 1 A 2 > A 3 and A 1, A 3 > 0;

6 110 Cristina Bercia, Romeo Bercia The third component of E 3 has to e strictly positive, or equivalently x 3 e x 3 > 0. Then we notice that A 1 > 0 if x 3 < rk (r H) + Kr. (10) This implies x 3 < 1 and since H > µr, we have A 3 > 0. With simple algera, we deduce the following Proposition 2.4. If E 3 exists, the condition (10) holds and ( H r (x 3e x 3 ) 1 KH ) r H e x 3 (1 x 3 ) ( x3 e x 3 ( 1 + KH ) r r H e x 3 (1 x 3 ) K ) r H e 2x 3 x 3 (1 x 3 ) (11) ( ) +(x 3 e x 3 ) µke x 3 H(r H) (1 x 3 ) (x r 2 3 e x 3 ) > 0 then the equilirium E 3 is locally asymptotically stale. The previous propositions are sintethized in the following Theorem 2.1. Let > 1, H 1 < H 0 and e < < e. Then: i) If H 2 > µ, for H (µ, H 2 ), the system (2) has at least three equiliria: E 0 as attractive node, E 1 and E 2 hyperolic saddles; ii) For max(µ, H 2 ) < H < H 1, the system has the equilirium points E 0, E 1, E 2 as in the previous case and E 3 which is locally asymptotically stale (l.a.s.) if conditions (10) and (11) hold; iii) For H = H 2, E 2 and E 3 concide; iv) For H = H 1, E 1 and E 3 concide; v) For H 1 < H < r, E 3 may e unphysical, E 1 ecomes l.a.s., E 0 remains attractive node and E 2 hyperolic saddle. Theorem 2.2. Let H 1 < H 0 and > e. Then: i) For µ < H < H 1, the system has the equilirium points E 0, E 1 hyperolic saddles and E 3 which is l.a.s. if conditions (10) and (11) hold; ii) For H = H 1, E 1 and E 3 concide; iii) For H 1 < H < r, E 3 may e unphysical, E 1 ecomes l.a.s., E 0 remains hyperolic saddle. 3. Bifurcation analysis First we take as the control parameter for the system (2). Theorem 3.1. If H er( 1), X e( 1)+K 0 > 1, for = ed 1, the oundary equiliria E 1 and E 2 appear through a saddle-node ifurcation. For < e, these two equiliria do not exist.

7 Bifurcation Analysis of a Model of Three-level Food Chain in a Mangrove Ecosystem 111 Proof. We fix = e, then x 1 = x 2 = 1 < X 0 and y1 = y2, so E 1 = E 2. The eigenvalues corresponding to E 1 are λ 1 = er( 1) H, λ e( 1)+K 2 = 0, λ 3 = 1, so we need λ 1 0 in order to have the codimension 1 ifurcation of saddle-node type. Let e DF Dv (E 1, e ) the Jacoian matrix of the system written in the form v = F (v, ), with the ifurcation parameter and v = (x, y, z), evaluated in E 1 for = e. We use a theorem (Sotomayor [2]) which gives necessary and sufficient conditions for a saddle-node ifurcation. These are: (SN1) DF Dv (E 1, e ) has a single zero eigenvalue which is fulfilled, due to the hypothesis of the theorem. Let e u = (1, e, 0) T its right eigenvector and w = (0, r, 1) T its left eigenvector. (SN2) w F (E 1, e ) = r( 1) 0. (SN3) w D vv F (E 1, e )(u, u) = 2 e ( 1)r 0. The nonzero conditions (SN2) and (SN3) imply that SN = { = e, H er( 1), X e( 1)+K 0 > 1 } is a saddle-node ifurcation surface in the parameters space. When the parameters pass from one side of the surface to the other side, the numer of the oundary equiliria changes from zero when < e to two hyperolic equiliria E 1,2 when > e, in the neighorhood of = e. These equiliria are connected y an orit that is asymptotic to E 1 for t and to E 2 for t. Now we take H as a control parameter for the system. Theorem 3.2. If > 1, > e and µ > K ( x 1) e 2, (12) ( x 1) + K then a) the equiliria E 1 and E 3 coincide for H = r( x 1 ) =: H K +( x 1 ) 1 at a point of transcritical ifurcation; ) E 2 and E 3 coincide for H = r( x 2 ) =: H K +( x 2 ) 2 also at a point of transcritical ifurcation, if, in addition < e. Proof. a) i) We fix H = H 1. The condition (12) is equivalent to H 1 < r Kµ+e2 K+e 2 which (see Proposition 2.2 and corrolary) implies that the solution x 3 of the equation (4), g(x) = 0, exists and it is unique. Then H = H 1 x 1 = KH. But (r H) x 1 < 1 and verifies xe x =, in consequence x 1e x 1 + x 1 = µrk r H 1 and x 1 = x 3. Also y 3 = KH 1 r H 1 = y1, z 3 = 0 K H 1 µr r H 1 which imply E 1 = E 3. From the proof of Proposition 2.3, we have: ii) For H = H 1, the eigenvalues for E 1 are λ 1 = 0, Re(λ 2,3 ) < 0, so only one eigenvalue is zero. iii) On the other hand, the equilirium E 1, from unstale (when H < H 1 ) ecomes stale for H > H 1. iv) We evaluate the matrix ( DF Dv F H ) at the ifurcation point (E1, H 1 ) and we

8 112 Cristina Bercia, Romeo Bercia find that F = (0, 0, H 0)T. It implies rank ( DF ) F Dv H = 2. From i)-iv) (see [5]), we deduce that in R 3 + {H > 0} the ranches of equiliria E 1 and E 3 intersect through a transcritical ifurcation at H = H 1 and change their stale manifold when we pass the critical parameter value. ) There are similar arguments. The condition < e ensures the existence of the equilirium E 2. Then H = H 2 x 3 = x 2 > 1. Also only one eigenvalue (see proof of Proposition 2.3) corresponding to E 2 is zero for H = H 1. These values of the parameters H = H 1, H = H 2, = e delimitate strata on parameters space induced y topological equivalence of the phase portraits. We are now investigating dynamic ifurcations. The equilirium E 3 is the only one which may experience Hopf ifurcation ecause only the interior equilirium can have a pair of purely imaginary eigenvalues. This necessary condition for E 3 to undergo a Hopf ifurcation is equivalent to A 1 A 2 = A 3, A 2 > 0 (13) (with A i given y (9)) together with the sufficient conditions for the existence of the interior equilirium (see Proposition 2.2). Also we need that the third eigenvalue of E 3 to e non-zero, i.e. A 3 0, which is satisfied if x 3 < 1 with g(x 3 ) = 0. In this case, we have A 3 > 0 and so the characteristic equation (8) admits the roots λ 1,2 = ±iω, ω > 0 and λ 3 = A 1 < Numerical results We take H as a control parameter. We fix the parameters in order to e e in the case e < < D 1 (see Theorem 2.1): K = 1; = 1.2; r = 0.3; = 0.66; = 0.2; µ = 0.1; = 1. This case corresponds with the existence of two oundary equiliria E i (x i, yi, 0), i = 1, 2. With a program in MAPLE, LamertW (0, D1 ) = we find the solutions of the equation xe x =, x 1 = ,x 2 = LamertW ( 1, ) = , then y 1 = , y2 = The static ifurcation parameters are H 2 = < H 1 = We solve numerically the system A 2 > 0, x 3 < 1, x 3e x 3 > 0 A 1 A 2 = A 3 g(x 3 ) = 0 (14) which are the necessary conditions for Hopf ifurcation of E 3 and we find the ifurcation parameter value H = H cr = and x 3 = H cr (H 2, H 1 ), H 1 < H 0 and we are in the hypothesis of Proposition 2.2 and its corollary. There is only one interior equilirium for any H (H 2, H 1 ).

9 Bifurcation Analysis of a Model of Three-level Food Chain in a Mangrove Ecosystem 113 Then, we investigate the appearance of a limit cycle when H is in the neighorhood of the critical parameter value. We find that for H < H cr an asymptotically stale limit cycle appears. For example, when H = 0.11 (H 2, H cr ), the solution of equation g(x 3 ) = 0, x 3 e x 3 > is x 3 = Then y 3 = ; z 3 = With initial conditions close to E 3, we integrate numerically the system. With a program in MATLAB we otain the phase portrait. The trajectories come together close to the saddle connection etween E 1 and E 2, towards E 1 and then tend to a stale limit cycle, while E 3 is repelling. (see Figure 1) Figure 1. Left: A stale limit cycle when K = 1; = 1.2; r = 0.3; = 0.66; = 0.2; µ = 0.1; = 1; H = 0.11 < H cr. Right: Time oscillations of populations corresponding to one of the trajectories, with initial values (0.6228; ; ). Figure 2. Left: Trajectories tending to the attractive focus E 3. The modified parameter is H = 0.13 > H cr. Right: Time evolution of the three populations corresponding to one of the trajectories, with initial values (0.7666; ; ).

10 114 Cristina Bercia, Romeo Bercia For H = 0.13 (H cr, H 1 ), trajectories initiated in a small neighorhood of E 3, come together very close to the saddle connection E 2 E 1 and tend for t to E 3 as a focus. (see Figure 2) In consequence, (E 3, H cr ) is a point of supercritical Hopf ifurcation ecause, when H varies and passes the critical value, from a stale equilirium E 3 for H > H cr, it appears a stale limit cycle for H < H cr, while E 3 loses its staility. 5. Conclusions The harvesting rate (H) of top predator can control the stale equilirium point or periodic state of coexistence of the three species. First of all, the model illustrates the phenomenon of iological overharvesting when H > H 1 and in this case the top-predator goes to extinction. If the detritus-detritivores conversion rate is not high, e<< e, the scenario of extinction of oth detritivores and top-predator is possile, for any value of the harvesting rate, depending on initial population levels. When we decrease H, for H (H cr, H 1 ), all the populations coexist in a form of a stale equilirium, under certain initial conditions. When H passes through a critical value, H cr, the system undergoes a Hopf ifurcation, namely for H (max(µ, H 2 ), H cr ), different populations of the system will start to oscillate with a finite period around the equilirium point of coexistence. Since H = d 2+h, it turns out that also for increasing large values of a washout rate (a) of detritus, the system changes from a stale state to an unstale state and the populations will survive through periodic fluctuations. Mathematically, we would like to point out here that our analysis of the model is a first look at the local ifurcations, ut it is not complete. For example a study of codimension 2 ifurcations will reveal richer dynamics. R E F E R E N C E S [1] H. I. Freedman, G. S. K. Wolkowicz, Predator-prey systems with group defence: The paradox of enrichment revisited, Bull. Math. Biol., 48(1986), [2] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag New York, 1983 [3] B. Mukhopadhyay, R. Bhattacharyya, Modelling Phytoplankton Allelopathy in a Nutrient-Plankton Model with Spatial Heterogeneity, Ecological Modelling, 198(2006), 198, p [4] B. Mukhopadhyay, R. Bhattacharyya, Diffusion Induced Shift of Bifurcation Point in a Mangrove Ecosystem Food-chain Model with Harvesting, Natural Resource Modeling, 22(2009), No 3, p [5] A. H. Nayfeh, B. Balachandran, Applied Nonlinear Dynamics, Analytical, Computational and Experimental Methods, Wiley Series, [6] S. Ruan, H. I. Freedman, Persistence in the three species food chain models with group defence, Math. Biosci., 107(1991), p