Period Doubling Cascades in a Predator-Prey Model with a Scavenger

Transcription

1 Period Doubling Cascades in a Predator-Prey Model with a Scavenger Joseph P. Previte Kathleen A. Hoffman June 14, 2012 Abstract The dynamics of the classic planar two-species Lotka-Volterra predator-prey model are well understood. We introduce a scavenger species, who scavenges the predator and is also a predator of the common prey. For this model, we analytically prove that all trajectories are bounded in forward time, and numerically demonstrate persistent bounded paired cascades of period-doubling orbits over a wide range of parameter values. Standard analytical and numerical techniques are used in the analysis of this model making it an ideal pedagogical tool. We include exercises and an open-ended project to promote mastery of these techniques. 1 Introduction Pioneering work by Lotka [36, 35] and Volterra [56] successfully captured the oscillations in populations of a predator and its prey. The classical set of data that span almost a century is the Canadian lynx and snowshoe hare pelt-trading records of the Hudson Bay Company [40]. The Lotka-Volterra equations are a cornerstone in the field of mathematical ecology and a significant amount of literature devoted to studying variants of these equations has been established. Our goal is to introduce a third scavenger species to the classical predator-prey equations in a biologically reasonable way. We characterize the third species as a scavenger who is also a predator of the prey and scavenges the carcasses of the predator (a scavenger has no negative effects on the population that it scavenges). A possible triple of such species are hyena/lion/antelope, where the hyena scavenges lion carcasses and preys upon antelope. The novel aspects of this model together with its accessibility make it ideal for inclusion in a senior-level or introductory graduate-level continuous dynamical systems or ordinary differential equations course. The analysis involves standard analytical Mathematics Subject Classifications: 92D25, 92H40, This work was partially supported by NSF-DMS-# , NSF-DMS-# and NSF-DMS- # School of Science, Penn State Erie, The Behrend College, Erie, PA Department of Mathematics and Statistics, UMBC, Baltimore MD

2 and numerical techniques, lending itself as an ideal pedagogical tool. We assume the reader has an understanding of linearization, Hopf bifurcations, continuity with respect to initial conditions, and elementary numerical techniques. The reader will be given ample opportunity to construct and use Lyapunov functions (which is a bit of an art form), develop trapping regions for solutions to ordinary differential equations, and use advanced techniques such as the Routh-Hurwitz test to show the existence of a Hopf bifurcation. Additionally, we have included many biologically motivated exercises throughout the paper whose solutions involve both analytical and numerical techniques. Our focus is on Lotka-Volterra predation equations in which the predator benefits from interaction with the prey, in contrast with the Lotka-Volterra competition equations in which no species benefits from the presence of other species [13]. For literature on Lotka-Volterra competition models, see for example, [3, 4, 5, 23, 24, 25, 52, 8, 59]. General Lotka-Volterra predation systems were studied in many contexts by many different authors. We do not attempt to review all the literature, but instead point to a few specific examples. One variation of the Lotka-Volterra predation equations is to consider density dependent prey (see Hirsch [21] and references therein, as well as Pielou [41], Rosenzweig [45], and Schoener [51]). Another variation is to include varying attack rates, as was considered by Ivlev [30], Holling [26], Rosenzweig [45], and Takahashi [53], for example. Kuznetsov [33] studied global bifurcations in tritrophic food chains for the Rosenzweig-MacArthur model and constructed organizing centers of overall bifurcations. Studies on three-dimensional predation food chains include [19, 9], who show a teacup-shaped attractor for a three-species Lotka-Volterra predation model with a type II functional response. Muratori et al [38, 39] explored a three-species Lotka-Volterra predation model motivated by insects preying on trees. Gilpin [15] showed the existence of spiral dynamics for a two prey and one predator system. His work formed the basis for other studies [19, 32, 49, 50, 55], some of which demonstrate the existence of unbounded period doubling cascades. Perhaps the work closest in nature to the one presented here can be found in [54]. Tanabe and Namba [54] numerically demonstrate that the addition of an omnivore (defined as feeding on more than one trophic level [42, 43]) leads to a Hopf bifurcation and period doubling cascades. We consider a slightly different model than [54]. In our model, we analytically establish all solutions with positive initial conditions are bounded in forward time, which is not the case in the model of [54]. Additionally, in contrast to the unbounded cascades seen in [54], our model gives rise to bounded paired cascades, recently defined in [48, 47], and is one of the first nontrivial biological systems that demonstrate the existence of these types of cascades. The paper is organized as follows. We begin with a review of the dynamics of the classic two-species Lotka-Volterra predator-prey model in section 2. We then add a scavenger to the model in section 3, and discuss global stability of trajectories in each of the coordinate planes (the dynamics of any two species in the absence of the third) in section 3.1. A local analysis of the three-dimensional equilibria can be found in section 3.2, and in section 3.3, we show that all orbits with positive initial conditions remain bounded in forward time. In section 3.4, we classify the omega-limit set of all trajectories with positive initial conditions when there is no equilibrium point in the open first octant. Section 3.5 contains a description of the Hopf bifurcation with nu- 2

3 merically computed bifurcation diagrams for selected parameter values. Additionally, a bounded paired cascade with a period three orbit is shown. Finally, a summary is provided in section 4. 2 Review of Analysis of the Predator Prey Model 2.1 The Planar Lotka-Volterra System In this section, we review well-known results of the planar Lotka-Volterra model. The differential equations that model the population dynamics of a predator Y and its prey X with a carrying capacity, the maximum population size that is sustainable due to limited environmental resources, are: where, A is the net growth rate of X dx dτ = X(A BX CY ), dy dτ = Y ( D + EX), B is such that A/B is the carrying capacity of X C is the rate of change of the prey population in response to the presence of a predator D is the natural death rate of the predator in the absence of prey E is the rate of change of the predator population in response to the presence of prey All parameters are positive, and all variables non-negative. In the classical Lotka Volterra model, it is customary to take B = 0 as the trajectories with positive initial conditions are bounded, but in order to maintain bounded solutions in models where additional species are added requires a logistic terms on X [6]. This quadratic term in X corresponds to the effective carrying capacity of the prey population. We also mention that different functional responses (terms representing the interactions from predation) have been proposed and studied. In this paper, we will utilize the simplest of functional responses, Type I functional responses which are products of the interacting species. In order to work with the fewest number of parameters, we first consider the change of variable t = Aτ, so that dx = dx dτ dτ = 1 dx A dτ and we obtain: dx = X(1 B A X C A Y ), dy = Y ( D A + E A X). Next, consider the changes of variables x = E A X and y = C A Y, which gives dx = E dx A = x(1 B E x y) 3 (1)

4 and dy = C dy A = y( D A + x). Relabeling the constants b = B E and c = D A, we arrive at dx = x(1 bx y), dy = y( c + x). The behavior of solutions of (2) reflect the behavior of solutions to (1) by simply reversing the coordinate change. We further note that the planes x = 0 and y = 0 are invariant with respect to the flow of (2). 2.2 Analysis of Equilibria Equilibria are solutions that are found by setting the derivatives equal to zero. For this system there are generically (bc 1) three equilibria for the planar system (0, 0), (1/b, 0), (c, 1 bc). Local stability of an equilibrium can be determined by the signs of the real parts of the eigenvalues of the Jacobian evaluated at the equilibrium. A. (0, 0): For all b and c, the origin (0, 0) is a saddle, i.e., the Jacobian has one positive and one negative eigenvalue, with the y-axis as the stable manifold and the x-axis as the unstable manifold. In general, the stable and unstable manifolds are tangent to the eigenvectors associated with the eigenvalues whose real parts are negative and positive, respectively. This equilibrium corresponds to extinction of both species. B. (1/b, 0): This equilibrium point represents the limiting population of the prey as 1/b in the absence of the predator. This equilibrium is a stable node for 1 bc < 0, a saddle for 1 bc > 0 with the positive x-axis as the stable manifold, and a singular stable node for bc = 1. C. (c, 1 bc): A nontrivial equilibrium (c, 1 bc) corresponds to a stable equilibrium for 1 bc > 0, and coincides with equilibrium B when bc = 1. For bc < 4 b+4 it is a stable spiral, otherwise, it is a stable node in the open first quadrant. When 1 bc < 0, this equilibrium is not in the first quadrant, and is of no biological relevance. 2.3 Global Analysis An equilibrium is locally asymptotically stable if solutions with initial conditions sufficiently close to the equilibrium are attracted to the equilibrium in forward time. If the real parts of the eigenvalues of the Jacobian evaluated at the equilibrium point are all negative then the equilibrium point is locally stable. Global stability, on the other hand, is concerned with whether initial conditions far from the equilibrium are also attracted to the equilibrium in forward time. For global stability analysis, we use a Lyapunov-like function. For a differential equation or system of differential equations, (2) 4

5 L is called a Lyapunov function if it is a differentiable function defined in a neighborhood of a equilibrium point P with an absolute minimum at P with the property that for any solution with initial conditions not equal to P, dl < 0. The existence of such a function implies that the equilibrium point is asymptotically stable for all solutions with initial conditions in the domain of L (a more complete discussion of Lyapunov functions can by found in a dynamical systems textbook, such as [17, p.4], [22, p.194], [37, p. 123]). The reader should be aware that the use of the term Lyapunov function is not consistent throughout the literature. We consider Lyapunov functions for equations of the form (2). We note that trajectories for the model with b = 0 are given by the curves x c ln x + y ln y = K. For b 0, this motivates us to investigate functions of the type f(x, y) = C 1 x C 2 ln x + C 3 y C 4 ln y, (3) where C i 0 as possible Lynapuov functions. Lyapunov functions of the form (3) have a long and venerable history. Volterra [56] first postulated the form of this function, however, Goh [16] used functions of this form to prove global asymptotic stability of the nontrivial equilibrium for a food chain with n species with a carrying capacity added to the prey equations. Under appropriate assumptions Harrison [18] showed that the entire positive orthant is in the domain of attraction of this nontrivial equilibrium. Also, see [31, 28, 1, 57], for example, for other sources on the form of this Lyapunov function in predator-prey systems. Next, we demonstrate how to find appropriate constants in (3). Standard calculus techniques show that the function g(w) = w K ln w has an absolute minimum at w = K. Hence, in order for f(x, y) to have a minimum at (c, 1 bc), we require C 2 = cc 1 and C 4 = (1 bc)c 3. Since a non-zero constant multiple of a Lyapunov function is also a Lyapunov function, we may take C 1 = 1. Lastly, since we wish d f(x(t), y(t)) < 0 for all solutions (x(t), y(t)) with positive initial conditions, we compute = Choosing C 3 = 1, leads to d f(x(t), y(t)) = f dx x + f dy y ( 1 c ) ( x(1 bx y) + C bc x y ) y( c + x). d f(x(t), y(t)) = b(x c)2 0. (4) Therefore, we consider the function f(x, y) = x c ln x + y (1 bc) ln y. Theorem 2.1. Assume 1 bc > 0 and consider the function f(x, y) = x c ln x + y (1 bc) ln y. 5

6 This function has one absolute minimum at (c, 1 bc) and for any solution (x(t), y(t)) with positive initial conditions to system (2), we have lim f(x(t), y(t)) = f(c, 1 bc) t which implies that all solutions with positive initial conditions limit to (c, 1 bc) in forward time. Proof. As mentioned above, f(x, y) has a unique absolute minimum at (c, 1 bc). For any solution (x(t), y(t)) with positive initial conditions to system (2), we have d f(x(t), y(t)) = b(x c)2 0, from (4). (If not for the exception x = c, f(x, y) would be a true Lyapunov function). Note that for any solution (x(t), y(t)) with positive initial condition that is not the equilibrium point (c, 1 bc), there can only be isolated t values where x(t) = c, for when x = c we have dx 0. This implies (by continuity of f) that the function f(x(t), y(t)) is strictly decreasing for any solution (x(t), y(t)) with initial conditions in the open first quadrant not equal to (c, 1 bc). Since f is continuous, bounded below, and decreasing lim t f(x(t), y(t)) exists. By continuity of f and continuity of initial conditions, any point in the omega limit set of this trajectory (The omega limit set is nonempty since one can use f itself to show that forward trajectories are bounded.) must a have constant value of f for all t. This implies that the omega limit set can only be (c, 1 bc). Thus, lim t x(t) = c. and lim t y(t) = (1 bc). Theorem 2.2. If 1 bc 0, then all solutions with positive initial conditions limit to ( 1 b, 0) in forward time. Proof. Compose any solution (x(t), y(t)) with positive initial conditions with the function F (x, y) = x 1 b y. Then df = x 1 b y ( 1 b x 1 b y + ( c + x)) = x 1 b y ( 1 b c 1 b y), which is strictly negative for all x, y > 0. So all solutions with positive initial conditions must limit to the union of the axes. All solutions with initial conditions on the positive axes tend to either (0, 0) (a saddle whose stable manifold is the positive y axis) or ( 1 b, 0) (a stable node). Thus, by continuity of initial conditions, all orbits with initial conditions in the open first quadrant must limit to ( 1 b, 0) Finding the right function as in the previous proof is somewhat of an art form, however, for equations of this type, functions of the form x α y β are often useful in showing that solutions with positive initial conditions limit to the coordinate axes. This idea will be used throughout this manuscript. Exercise 2.3. The argument in the previous proof is not unique. We chose to include this form because it will be used later. In this exercise, instead, consider a function of the form g(x, y) = c 1 x + c 2 ln x + c 3 y + c 4 ln y. Find the constants c i such that dg is strictly negative for all positive values of x and y. This will imply that solutions limit to equilibrium point (1/b, 0) of equation (2) for 1 bc 0. In summary, if 1 bc 0, then all trajectories with x(0) > 0 and y(0) > 0 limit to (1/b, 0), a stable node, and the system cannot sustain the predator population. The predator population goes extinct and the prey population limits to its carrying capacity. If 1 bc > 0, the populations approach (c, 1 bc), and both species persist. 6

7 3 A Scavenger Model Many studies have shown that omnivory (feeding on more than one trophic level) is commonly found in food chains [7, 10, 2, 46] and a review by Devault et al. [11], indicates terrestrial scavenging has been underestimated by the ecological community [20, 44, 27, 58]. In this section, we introduce a third species z which is a predator of the prey x, and consumes the carcasses of the predator y, but has no direct inhibitory effects on the population of the predator. In [6], a scavenger species was studied where the scavenger had no effect on the predator or the prey. Thus, the predator and prey were independent of the scavenger. In this study, the scavenger species scavenges the predator, while also being a predator of the common prey species. We consider the following general three-species model: dx dτ = X(A BX CY DZ), dy dτ = Y ( E + F X), (5) dz dτ = Z( G + HX + IY JZ), where X is the prey, Y is the predator, and Z scavenges Y and preys upon X. The parameters can be interpreted as follows: A is the net growth rate associated with X B and J are implicitly related to the carrying capacities of X and Z, respectively C and D are the rates of change of the prey population in response to the presence of Y and Z, respectively E and G are the natural death rate of Y and Z in the absence of prey F and H are the rates of change in the predator and scavenger populations, respectively, in the presence of the prey X I is the rate of change in the scavenger population Z in the presence of the population Y All parameters are assumed to be positive, and all variables non-negative. Exercise 3.1. Show that system (5) can be transformed into dx = x(1 y z bx), dy = y( c + x), dz = z( e + fx + gy hz), after four transformations. List the new variables x, y, z, and t and parameters b, c, e, f, g and h in terms of the variables and parameters in system (5). (6) 7

8 3.1 Analysis of Coordinate Planes The coordinate planes x = 0, y = 0, and z = 0 are invariant, since if x = 0, then so is ẋ, and similarly for y and z. The dynamics between x and y in the absence of z are as described in section 2. Consider the x = 0 plane, or the dynamics of the predator and the scavenger, in the absence of the prey. The dynamics are given by the two-dimensional system dy = cy, dz = z( e + gy hz). Exercise 3.2. Show that the origin is asymptotically stable and that it is the only equilibrium point in the first quadrant. Use a function of the type F (y, z) = y β z γ for an appropriate choice of β and γ to show that any solution with initial condition y(0) 0 and z(0) 0 will approach the origin (similar to the argument given in the proof of Theorem 2.2). In the y = 0 plane, where the scavenger z preys upon x and both x and z are limited by carrying capacity, the dynamics are given by a two-dimensional equation: dx = x(1 z bx), dz = z( e + fx hz). This is a predator-prey system with a carrying capacity on both the predator z and the prey x whose analysis is similar to that of the predator-prey system in 2.1. Exercise 3.3. a. Show for f be 0, all trajectories with positive initial conditions limit to the equilibrium point ( 1 b, 0). b. Show for f be > 0 there exists an interior equilibrium point (ˆx, ẑ). Further, use a function of the form F (x, z) = C 1 x C 2 ln x + C 3 z C 4 ln z and an argument similar to the proof of Theorem 2.1 to show that any solution (x(t), z(t)) with positive initial conditions limits to that interior equilibrium (ˆx, ẑ). We conclude by summarizing: the coordinate planes are invariant, there are no periodic solutions on the positive coordinate planes, and all trajectories with positive initial conditions on those planes approach equilibria. 3.2 Analysis of Equilibria for the Three-Species Model The three equilibria from the planar system (2) persist in the three dimensional system with the ( third coordinate ) set to zero: (0, 0, 0), (1/b, 0, 0), (c, 1 bc, 0), and the equilibrium h+e be+f bh+f, 0, bh+f persists with the second coordinate set to zero. Additionally, there is an interior equilibrium: ( ) h + e fc bch e + fc + g gbc (x, y, z ) = c,,. g + h g + h 8 (7) (8)

9 Figure 1: Equilibria: For fixed g = 13, b = 0.9, c = 0.1, h = 15, e = 11.32, f = 0.1, the equilibria are plotted with crossed circles and are all unstable. For these parameter values (0, 0, 0) is a saddle with stable manifold the yz plane. A trajectory in the yz plane limiting to (0, 0, 0) is shown. The equibrium point (1/b, 0, 0) = ( , 0, 0) is a saddle with stable manifold the xz plane. A trajectory in the xz plane limiting to (1/b, 0, 0) is shown. The equilibrium point (c, 1 bc, 0) = (0.1, 0.91, 0) is saddle with stable manifold the xy plane (it is a stable spiral) a limiting trajectory shown. The in- ( terior equilibrium, given by c, h+e fc bch g+h, e+fc+g gbc g+h ) = (0.1, , ), is unstable. Note that equilibria D does not live in the positive octant for these parameter values. Also illustrated by the bold closed curve is a stable periodic orbit with initial conditions ( , , ). 9

10 Figure 1 illustrates the equilibria for particular values of the parameters and a stable periodic orbit shown in bold. Exercise 3.4. Use the eigenvalues of the Jacobian to show that: A. (0, 0, 0) is a saddle with stable manifold the yz plane and unstable manifold the x axis. B. (1/b, 0, 0) is a stable node for be > f and 1 bc < 0, otherwise it is a saddle. C. (c, 1 bc, 0) is stable if and only if e + fc + g gbc < 0. For bc < 4 b+4 it is a stable spiral in the xy plane and otherwise a stable node in the xy plane. ( ) h+e be+f h+e D. bh+f, 0, bh+f is stable if and only if bh+f < c. In the event that h+e bh+f ( ) > c but h+e be+f f be > 0, then bh+f, 0, bh+f is a saddle with stable manifold the positive xz plane. The analysis of the interior equilibrium point is a bit more complicated. ( ) E. c, h+e fc bch g+h, e+fc+g gbc g+h The Jacobian evaluated at the above equilibrium point has the form: bx x x y 0 0 z f z g hz where, (x, y, z ) = (c, h+e fc bch g+h, e+fc+g gbc g+h ) and has characteristic polynomial λ 3 + (hz + bx )λ 2 + x (bhz + y + fz )λ + x y z (g + h). The eigenvalues of the Jacobian are sufficiently messy as to require us to rely on the Routh-Hurwitz method. The Routh-Hurwitz Test applied to a general third degree polynomial a 3 λ 3 + a 2 λ 2 + a 1 λ + a 0 effectively states the number of sign changes in the sequence {a 3, a 2, H, a 0 } where H = a 2 a 1 a 3 a 0, is equal to the number of roots of the polynomial having positive real part and if all entries in the sequence are non-zero and of the same sign, then all roots have negative real part. (Note: We are only stating this criterion as it applies to cubic polynomials, the Routh-Hurwitz criterion can be applied to any degree polynomial, see [29]). For our characteristic polynomial, a 3, a 2 and a 0 are clearly positive. We conclude that if H = x (hz + bx )(bhz + y + fz ) x y z (g + h) > 0 then the equilibrium point (x, y, z ) is stable, and if H < 0 then the equilibrium point is unstable. Note that if H = 0, then the characteristic polynomial becomes λ 3 + a 2 λ 2 + a 1 λ + a 2 a 1 = (λ + a 2 )(λ 2 + a 1 ) = 0, which has two purely imaginary roots. In particular, if a path is taken in parameter space γ(s) = (b(s), c(s), e(s), f(s), g(s), h(s)) with H(γ(0)) = 0 and 10

11 Figure 2: Hopf curve: For fixed b = 0.9, c = 0.1, f = 0.1, g = 13, the curve illustrated in the h-e plane demonstrates how the Hopf bifurcation changes as the parameters e and h vary. This figure illustrates the range of values of h for which there exists Hopf bifurcations for two distinct positive values of e. d ds H(γ(s)) s=0 0 then a Hopf bifurcation occurs at the parameter γ(0). Note that when g = 0, H > 0, so no Hopf bifurcations occur in this case. In Figure 2, for the values b = 0.9, c = 0.1, f = 0.1, g = 13, we plot H = x (hz + bx )(bhz + y + fz ) x y z (g + h) = 0 as an implicit function of e and h. Note that for a range of positive values of h, there are two values of e for which H = 0 (i.e. a range of horizontal lines that intersect the curve twice with positive values of the parameters). We note that inside the curve in figure 2, we have H < 0, and thus unstable equilibria, and outside the curve H > 0, and thus stable equilibria. In later sections, this figure will motivate us to use these two parameters as bifurcation parameters. We will demonstrate a parameter regime where a bounded cascade of solutions exist for parameter values inside the closed curve in Figure Bounded Orbits In this section, we show that the trajectories with positive initial conditions of (6) remain bounded. To this end, we define a trapping region bounded by the coordinate axes such that the vector field points toward the interior of the region. Thus, trajectories that start in that region remain in that region for all forward time. We then show that for all positive initial conditions, trajectories must eventually enter a bounded region, and hence conclude that orbits are bounded in forward time. The analysis of the bounded orbits naturally breaks into two cases. In the case 1 bc 0, one can use a function of the type used in Theorem 2.2 to show that 11

12 trajectories with positive initial conditions approach the union of the x and y axes and obtain the following result: Exercise 3.5. When 1 bc 0, all trajectories ( with) positive initial conditions limit to h+e be+f (1/b, 0, 0) when be + f 0 and to bh+f, 0, bh+f when be + f 0. (Hint: First use F (x, y) = x 1 b y to show that y 0, then use continuity of initial conditions and results from the analysis of the coordinate planes to deduce the desired result.) In the second case, 1 bc > 0, we will show that solutions get trapped in a bounded region. Lemma 3.6. In the case 1 bc > 0, any trajectory (x(t), y(t), z(t)) with positive initial conditions (x 0, y 0, z 0 ), enters a trapping region Q R (see Figure 3 for a twodimensional projection of Q R ) in finite time, which implies that trajectories are attracted to a bounded region. Figure 3: A trapping region: This figure depicts the two-dimensional projections of Q 1 (which is shaded) and Q R for R > 1 b b. The regions Q 1 and Q R are independent of b z. For any point on R > 1 b, the vector field associated with the differential equation points inward, making each Q R a trapping region where trajectories with positive initial conditions limit to Q 1 in forward time. b Proof. We first construct a function defined in the first open octant: 1 x b x, 0 < y 1 xy b c x, 1 y, F (x, y, z) = 1 b xyb cy b 0 < x c, b 1 bc y else 1 b 12

13 This function is defined and continuous for all (x, y, z) in the first octant (see Figure 3), moreover F (x, y, z) 1 b, by virtue of the way F is defined. For R 1 b, define Q R = {(x, y, z) : F (x, y, z) = R}. Since F (x, y, z) is independent of z, the sets Q R (see Figure 3) can be viewed in the plane by extension in the z-direction. Although F (x, y, z) is continuous on the entire first octant, it is not differentiable at the boundaries of its regions of definition. Let (x 0, y 0, z 0 ) be such that F (x 0, y 0, z 0 ) = R 0 > 1 b. Let (x(t), y(t), z(t)) be the solution with x(0) = x 0, y(0) = y 0, and z(0) = z 0. We will show that F (x(t), y(t), z(t)) is decreasing. First, consider where 0 < y 0 < 1 and x 0 1 df b and compute = dx = x(1 y z bx). Since x 0 1 df b, we have (x 0,y 0,z 0)< 0, which means the vector field points inward on this region of Q R0. Next, consider x 0 y0 b = R 0 1 b for x 0 > c and y 0 > 1 and compute df = d(xyb ) = xy b (1 y bc z) which, for y 0 > 1, is strictly negative. So, once again along this segment of Q R0, df (x0,y 0,z 0)< 0, and the vector field points into the region Q R0. The last segment of Q R0 is 0 < x 0 < c, y 0 b 1 where we again compute df b 1 dy = cby = bcy b ( x + c). Since x < c along this segment, we have df (x0,y 0,z 0)< 0, so again, the vector field points into the region. Finally, consider an initial condition on the boundaries of definition for F, that is, the corners where the segments intersect. There are two cases. In the first case, x 0 > 1 b and y 0 = 1 we have both dx < 0 and dy < 0 so we must have F (x(t), y(t), z(t)) is decreasing since both of its representations on this boundary have negative derivatives (recall F is continuous, but not differentiable). In the second case x 0 = c and y 0 > 1, note that dx < 0, which means that x(t) c except at t = 0. Therefore (by the previous analysis when x 0 0), F (x(t), y(t), z(t)) is decreasing on intervals of the form ( δ, 0) and (0, δ), which implies (by continuity) that F is decreasing on all of ( δ, δ). Since F (x(t), y(t), z(t)) is decreasing and continuous, one can show that for any ɛ > 0 and any initial condition (x 0, y 0, z 0 ) in the domain of F, there must exist T > 0 so that F (x(t), y(t), z(t)) 1 b + ɛ for all t T. One can explicitly (but not easily) determine such a value of T in terms of (x 0, y 0, z 0 ), the parameters, and ɛ. In particular, for ɛ = 1 b > 0, after a finite amount of time we may assume that x(t) 2 b and y(t) b 2 cb. Thus, after a finite amount of time, dz = z( e + fx + gy hz) z ( e + 2fb ) 2cb + g b hz ( ) which is clearly bounded away from zero for values of z > 1 2f h b + g b 2 cb. Therefore, all trajectories with positive initial conditions are attracted to the bounded region { ( (x, y, z) : F (x, y, z) 2 b and 0 z < 1 )} 2f 2 h b + g b. cb bc, The dimensionless parameter h is associated with the parameter J in (5), which was mentioned to be related to the carrying capacity of the population z. A bound 13

14 ( 2f b + g b 2 cb ). It is on the carrying capacity is given in the previous proof as z < 1 h apparent that as h 0, this bound on the carrying capacity goes to infinity. Thus, since z corresponds to the top predator, and experiences no predation, the linear death rate corresponding to the term with the parameter e is not sufficient to bound the orbits using this argument and the quadratic term is necessary. As a consequence of Exercise 3.5 and Lemma 3.6, we have: Theorem 3.7. All trajectories with positive initial conditions are bounded in forward time. 3.4 Summary of Limiting Behavior In the previous section, we showed that all trajectories with positive initial conditions are bounded in forward time, as would be expected biologically. In this section, we further clarify the forward time limiting behavior of trajectories, provided that the first octant does not contain an interior equilibrium point. As was mentioned previously, if the interior equilibrium point exists, then all the other equilibrium points are unstable and no trajectory approaches the coordinate planes. All orbits with positive initial conditions are bounded, but the dynamics of the trajectories exhibit complicated dynamical structures, such as Hopf bifurcations, bistability, and period doubling cascades (see section 3.5). In this case, the limiting dynamics cannot be classified analytically. If the interior equilibrium point does not exist, then all trajectories with positive initial conditions limit in forward time to the coordinate planes, and hence will limit to one of the equilibrium points (which we will demonstrate in this section). For 1 bc 0, trajectories limit to (1/b, 0, 0) in forward time when be + f 0 and to ( h+e bh+f, 0, be+f bh+f ) when be+f 0, as was shown in Lemma 3.5. If on the other hand, 1 bc > 0, then the limiting behavior depends on the sign of e + g + fc gbc or h + e fc bhc. Exercise 3.8. Assume that e + g + fc gbc < 0 and h + e fc bhc > 0. Use a function of the form G(x, y, z) = x g y bg f z to argue that for any positive initial condition, trajectories approach the coordinate planes and limit to (c, 1 bc, 0) in forward time. Exercise 3.9. Assume e + g + fc gbc > 0 and h + e fc bhc < 0. Use a function of the form H(x, y, ( z) = x h y bh+f z 1 ) to show that for any positive initial condition, h+e be+f trajectories limit to bh+f, 0, bh+f in forward time. Exercise Assume e + g + fc gbc = 0 and h + e fc bhc = 0. Show that the equilibria in exercises 3.8 and 3.9 are the same. Hint: Use h+e fc bhc = 0 to show that the first coordinates of the equilibrium points are equal. Then add e+g+fc gbc = 0 and h + e fc bhc = 0 to show that 1 bc = 0, that is, the second coordinates are the same. Finally, explain why (e + h)(1 bc) = 0. Then expand that expression and add it to h + e fc bhc = 0 to show that f = be, thus equating the third components of the equilibrium points. 14

15 The table below summarizes the limiting behavior of all trajectories with positive initial conditions when the first octant does not contain an interior equilibrium point: 1 bc P arameters Limiting EquilibriumP oint 1 bc 0 f be 0 (1/b, 0, 0) 1 bc 0 f be 0 ( h+e be+f bh+f, 0, bh+f ) 1 bc > 0 e + g + fc gbc 0 and h + e fc bhc 0 ( h+e be+f bh+f, 0, bh+f ) 1 bc > 0 e + g + fc gbc 0 and h + e fc bhc 0 (c, 1 bc, 0) Next we argue that this table consists of all possible parameter values. For 1 bc 0, either f be 0 or f be 0, which exhausts all possible parameter values. If we consider the case when 1 bc > 0, then the cases not presented in the table include when e + g + fc gbc and h + e fc bhc have the same sign. Consider the case 1 bc > 0 and both e + g + fc gbc 0 and h + e fc bhc 0. We argue by contradiction that this parameter combination cannot exist. Suppose it does. The sum of e + g + fc gbc 0 and h + e fc bhc 0 is (1 bc)(h + g). However, 1 bc > 0, and h and g are positive parameters, contradicting the non-positivity of the sum. The case 1 bc > 0 and both e + g + fc gbc > 0 and h + e fc bhc > 0 imply that the interior equilibrium is in the first open octant. Consequently, the table exhaustively lists all possible parameter values when the interior equilibrium point is not in the positive first octant. In summary, if equilibrium point E is not in the open first octant, then at least one of the populations become extinct. For instance, in the last line of the previous table, the scavenger population becomes extinct and the populations of the predator and prey limit to the a classical constant predator-prey solutions as discussed in section 2. Considering the middle two lines of the table, the predator becomes extinct, and the prey and the scavenger co-exist, limiting to constant populations. For trajectories that approach the equilibrium (1/b, 0, 0) in forward time, both the predator and scavenger will become extinct, and the prey population limits to 1/b. 3.5 Period Doubling of the Interior Equilibrium Point In the previous section, we classified the forward-time limiting behavior of all trajectories with positive initial condition in the case that the open first octant does not contain an interior equilibrium point. The dynamics of the trajectories when the open first octant contains an interior equilibrium point is much more complex. We investigate this behavior through a sequence of examples that demonstrate that first the interior equilibrium point goes through a sequence of Hopf bifurcations that eventually leads to a period doubling cascade. For the diagrams below, the parameter h is fixed and bifurcation diagrams are created by varying the parameter e. The choice of these bifurcation parameters was motivated by Figure 2. These bifurcation diagrams were generated using the software package XPPAUT [14, 12]. To begin, we fix the parameters b = 0.9, c = 0.1, f = 0.1, g = 13. Initially, for h > 19, the initial equilibrium point illustrated in figure 4(b) is stable throughout the entire range of e. When h 18.6 the interior equilibrium point, shown in Figure 15

16 4(b), experiences a supercritical Hopf bifurcation at e 11.25, at which point a stable limit cycle shown in 4(c) is born. That limit cycle persists for a small window of the parameter e until undergoing a second supercritical bifurcation, collapsing onto the equilibrium point. (Note that the values for h and e have been obtained from formula for H, see equilibrium E in section 3.2). As h decreases further, the system s dynamics become more complex (cf. Figure 5(a)). For h 15, as e varies, the system again experiences two sequential Hopf bifurcations, only this time one is subcritical (at approximately e 10.72) and one is supercritical (e 11.57), forming a cardioid shape. Figure 5(a) shows a small parameter window in e immediately preceding the subcritical Hopf bifurcation where both the stable equilibrium point and a stable limit cycle coexist exhibiting two simultaneous stable structures, or bistability. Exercise For b = 9 10, c = f = 1 10, g = 13 and h = 18.7 show that a Hopf bifurcations occur when e = , Show that both Hopf bifurcations are subcritical. In order to do this, one must use a Taylor approximation of the center manifold up to quadratic terms and then analyze the resulting system on the center manifold (see sections of Guckenheimer and Holmes [17] for more information). Exercise For b = 9 10, c = f = 1 10, g = 13 and h = 15 show that Hopf bifurcations occur when e = , Determine whether the Hopf bifurcations are subcritical or supercritical. The previous figures illustrate the existence of Hopf bifurcation points, which demonstrate the existence of periodic orbits. Next, we numerically search for a period doubling cascade using the following algorithm. For each fixed value of e, a certain trajectory was iterated numerically using the classic Runge-Kutta method after 7000 units of time. The last 10% of the trajectory was analyzed. Let z denote the value of the z-coordinate of the internal equilibrium point. Whenever the trajectory nontrivially crossed the plane z = z, in a prefered direction, its y coordinate is plotted for that value of e. Effectively, for each value of e, one plots the y coordinates of the intersection(s) of the half plane z = z, y > y and the omega limit set of the iterated trajectory. As the value of the parameter e increased, the last value of the trajectory from the previous iteration was used as an initial condition for the next trajectory. Figures 6-12 show the results of the above algorithm. Plots were created for approximately 1700 values of e for each fixed value of h. We used the classic forth order Runge-Kutta algorithm in C to numerically solve the differential equations and the images were generated using GNUPLOT. We benchmarked our results with the Runge-Kutta45 algorithm, and obtained virtually identical results, so we opted for the classic Runge-Kutta method in order to shorten computer run time. Figures 6 and 7 illustrate the onset of a period doubling cascade from the stable periodic orbit. The trajectory plots 6(b) and 7(b) show examples of stable periodic orbits for a particular value of e. Similarly, figures 8(a) and 9(a) show the successive period doubling bifurcations and the corresponding trajectories (figures 8(b) and 9(b)) with period four and eight. Figure 10 depicts a bifurcation diagram with a period six orbit and its corresponding trajectory. Eventually, a period 3 orbit appears when h = 4.08, which 16

17 (a) (b) (c) Figure 4: Bifurcation diagram: For g = 13, b = 0.9, c = 0.2, h = 18.6, f = 0.1, e = 11.22, Figure 4(a) illustrates a bifurcation diagram in e. The line that is alternately solid then dashed, then solid again indicates 17 the y value of the interior equilibrium point. When the line is solid, the equilibrium is stable and when dashed it is unstable. The dotted closed curve indicates the maximum and minimum y values of a stable periodic orbit. In 4(a), we see two supercritical Hopf bifurcations at e and The stable equilibrium shown in 4(b) corresponds to the system with parameter e = indicated as a vertical line in 4(a). The stable periodic orbit for e = is pictured in 4(c) as the ω-limit set of two trajectories, one spiraling out to the limit cycle and the other spiraling into it.

18 (a) (b) Figure 5: Bifurcation diagram: For h = 15, g = 13, b = 0.9, c = 0.1, f = 0.1. Figure 5(a) illustrates a bifurcation diagram in e. Similar to figure 4, the line corresponds to the y coordinate of the equilibrium, the dashed line indicating unstable with solid indicating stable. The larger open circles making up the cardioid represent the maximum and minimum y values of an unstable periodic orbit, while the smaller solid circles correspond to stable periodic orbits. Again, there are two Hopf bifurcations, one subcritical at e and one supercritical at e Note the small window in e where two stable structures coexist. The vertical line at e = 10.6 indicates that a stable equilibrium point coexists with a stable limit cycle. Figure 5(b) illustrates one trajectory that approaches the equilibrium point and another that goes to the stable limit cycle, hence exhibiting bistability for these parameters. 18

19 is shown in Figure 11. The appearance of the period three orbit implies the existence of an infinite number of other periodic orbits [34]. Here, the result of Li and Yorke [34] is actually applied to the Poincaré map of the periodic orbit (see Guckenheimer and Holmes [17]). Finally, figure 12 shows the bounded paired cascade and two attractors corresponding to e = 11 and e = 10. A movie illustrating the evolution of the cascade can be found at According to [48, 47] who studied routes to chaos in maps, there are two types of cascades. The classic period doubling cascade produced by the logistic map is called a solitary cascade, is not connected to another cascade, and the cascade persists for all parameters larger than a certain value. However, a bounded paired cascade exhibits classic period doubling behavior leading to chaos, but then as the parameter increases, instead of the dynamics becoming more complicated, there is a second cascade that provides a route away from chaos. Our model produces bounded paired cascades that are contained within a compact interval in parameter space. This represents a nontrivial biological example exhibiting this phenomenon. Exercise The model clearly exhibits bistability (two coexisting stable structures as shown in Figure 5(a)) for parameters b = 0.9, c = 0.1, f = 0.1, g = 13, and h = 15 and e varying from Numerically determine the basin of attraction for each stable structure, that is, find the set of initial conditions for fixed parameter values in this window that limit to each structure in forward time. Note that the basin of attraction of the equilibrium point is small and requires numerical precision. Exercise a. Consider alternative formulations in which the effects of a scavenger of the predator crowds but does not consume the prey. That is, the scavenger has no benefit from that interaction, but the prey is inhibited by the scavenger. A possible triple of such species are water scavenger beetle/trout/mayfly, where the scavenger beetle consumes trout carcasses and crowds mayfly larvae. In equation (6) this scenario would be equivalent to setting f = 0. Argue why orbits in the first octant remain bounded in forward time and explain how the parameter value at which the Hopf bifurcation occurs changes. Try to find parameters for which a bounded pair cascade exists. b. Explain the biological scenario when f < 0 and characterize the behavior of all trajectories with positive initial conditions when the open first octant does not contain an equilibrium point. Are all orbits in the first octant bounded in this case? Can you demonstrate a bounded pair cascade? Exercise The proof to theorem 3.6 suggests an upper bound on the carrying capacity of the third species z. Numerically compute long term behavior of species z for parameter values in the chaotic regime of the bounded cascade. Use these calculations to estimate the carrying capacity and comment on the optimality of the bound suggested in the proof of theorem

20 (a) Bifurcation Diagram (b) Trajectory Figure 6: Evolution of the Cascade: For h = 7.0, the bifurcation diagram is shown in 6(a). Figure 6(a) is effectively computed by intersecting the half plane z = z, y > y with the stable limit cycle for various values of e, where y and z are the coordinates of the fixed point. Figure 6(b) illustrates the stable limit cycle and an unstable interior fixed point for e = Figure 6(b) is a trajectory for one value of e. 20

21 (a) Bifurcation Diagram (b) Trajectory Figure 7: Evolution of the Cascade: For h = 4.7, the bifurcation diagram in 7(a) shows a period doubling bifurcation. In 7(b), a period two trajectory is shown for e = Note that for e = 11.0 figure 7(a) indicates two intersections of the stable limit cycle with the half plane z = z, y > y. 21

22 (a) Bifurcation Diagram (b) Trajectory Figure 8: Evolution of the Cascade: For h = 4.48, the bifurcation diagram in 8(a) shows a second period doubling bifurcation. In 8(b), a period four trajectory is shown for e =

23 (a) Bifurcation Diagram (b) Trajectory Figure 9: Evolution of the Cascade: For h = 4.45, the bifurcation diagram in 9(a) shows a third period doubling bifurcation. In 9(b), a period eight trajectory is shown for e =

24 (a) Bifurcation Diagram (b) Trajectory Figure 10: Evolution of the Cascade: For h = 4.35, the bifurcation diagram in 10(a) shows a period six orbit, illustrated in 10(b) for e =

25 (a) Bifurcation Diagram (b) Trajecotry Figure 11: Evolution of the Cascade: For h = 4.08, the bifurcation diagram in 11(a) shows a period three orbit, shown in 11(b) for e =

26 (a) Bifurcation Diagram (b) Trajectory (c) Trajectory 26 Figure 12: Evolution of the Cascade: For h = 4.07, additional period doubling cascades appear on the period three branch of the bifurcation diagram. Stable attractors are shown in 12(b) and 12(c) for e = 11 and e = 10, respectively.

27 4 Summary We have introduced and studied a three-species model with a scavenger added to a predator prey system with simple linear functional response. In addition to having the biologically relevant property of bounded trajectories for positive initial conditions this model exhibits Hopf bifurcations, bistability, and chaos indicating the complexity of the dynamics due to adding a scavenger to a classical predator prey system. This novel model can be used as a powerful pedagogical tool which can be incorporated into a senior or graduate level continuous dynamical systems or ordinary differential equations class, since the analysis of the model simply uses standard tools from dynamical systems theory. 5 Acknowledgements Much of the numerical work for this paper was accomplished with the help of undergraduates Malorie Winters, James Greene, and Ben Nolting who participated in the NSF REU Program in Mathematical Biology at Penn State Erie from Much thanks also to Dr. Michael Rutter, who co-directed the REU and provided helpful insights throughout this research. Lastly, we thank the reviewers for their extensive and insightful comments. References [1] R. Aiken and L. Lapidus. The stability in two species interactions. Int. J. Syst. Sci., 4: , [2] M. Arim and P. A. Marquet. Intraguild predation: A widespread interation related to species biology. Ecol. Let., 7: , [3] A. Arneodo, P. Coullet, J. Peyraud, and C. Tresser. Strange attractors in Volterra equations for species competition. J. Math. Biol., 14: , [4] A. Arneodo, P. Coullet, and C. Tresser. Occurrence of strange attractors in threedimensional Volterra equations. Phys. Let., 79A: , [5] G. J. Butler and P. Waltman. Bifurcation from a limit cycle in a two predator-one prey ecosystem modeled on a chemostat. J. Math. Biol., 12: , [6] E. Chauvet, J. Paullet, J. P. Previte, and Z. Walls. A Lotka-Volterra three species food chain. Math Magazine, 75: , [7] M. Coll and M. Guershon. Omnivory in terrestrial arthropods: Mixing plant and prey diets. Ann. Rev. Entomology, 47: , [8] F. Montes de Oca and M. L. Zeeman. Balancing survival and extinction in nonautonomous competitive Lotka-Volterra systems. J. Math. Anal. and its Appl., 192: ,

28 [9] B. Deng. Food chain chaos with canard explosion. Chaos, 14(4), [10] R. F. Denno and W. F. Fagan. Might nitrogen limitation promote omnivory among carnivorous arthropods? Ecol., 84: , [11] T. L. Devault, O. E. Rhodes, and J. A. Shivik. Scavenging by vertebrates: Behavioral, ecological, and evolutionary perspectives on an important energy transfer pathway in terrestrial systems. Oikos, 102: , [12] E. J. Doedel, H. B. Keller, and J. P. Kernévez. Numerical analysis and control of bifurcation problems, parts I and II. Int. J. of Bif. and Chaos, 3, 4: , , [13] L. Edelstein-Keshet. Mathematical Models in Biology, volume 46 of Classics in Applied Mathematics. SIAM, [14] B. Ermentrout. Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students. SIAM, [15] M. E. Gilpin. Spiral chaos in a predator-prey model. Am. Naturalist, 113: , [16] B. S. Goh. Global stability in many-species systems. Am. Nat., 11: , [17] J. Guckenheimer and P. Holmes. Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, volume 42 of Applied Mathematical Sciences. Springer- Verlag, [18] G. W. Harrison. Global stability of food chains. Am. Nat., 114(3): , [19] A. Hastings and T. Powell. Chaos in a three-species food chain. Ecol., 72: , [20] S. B. Heard. Processing chain ecology: Resource condition and interspecific interactions. J. Anim. Ecol., 63: , [21] B. T. Hirsch. Measuring marginal predation in animal groups. Behavioral Ecol., pages , [22] M. Hirsch, S. Smale, and R. Devaney. Differential Equations, Dynamical Systems and an Introduction to Chaos. Elsevier, [23] M. W. Hirsh. Systems of differential equations which are competitive or cooperative: III. Competing species. Nonlinearity, 1:51 71, [24] J. Hofbauer and K. Sigmund. Evolutionary game dynamics. Bull. of the AMS, 40: , [25] J. Hofbauer and J. So. Multiple limit cycles for three dimensional Lotka-Volterra equations. Appl. Math. Lett., 7(6):65 70,