Canards and Mixed-Mode Oscillations in a Singularly Perturbed Two Predators-One Prey Model Susmita Sadhu

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1 Proceedings of Dynamic Systems and Applications Canards and Mied-Mode Oscillations in a Singularly Perturbed Two Predators-One Prey Model Susmita Sadhu Department of Mathematics Georgia College & State University Milledgeville GA 306. ABSTRACT: We observe mied-mode oscillations in a two-trophic ecological model comprising of two predators competing for their common prey. Both predators ehibit Holling II functional response with one of the predators territorial having a density dependent mortality rate. Under the assumption that the growth rate of the prey is much faster than the growth rates of the predators the model can be transformed to a singular perturbed system with one fast and two slow variables. In the absence of the non-territorial predator we identify a canard eplosion in the subsystem by geometric analysis. The canard eplosion refers to a change from an outbreak dynamics to small oscillations around the two-species equilibrium state over an etremely narrow parameter interval. The full-system eperiences periodic outbreaks interspersed with crashes which are referred to as relaation oscillations and are commonly observed in systems with multiple time-scales. However more interestingly we observe mied-mode oscillations which are concatenations of small amplitude and large amplitude oscillations that indicate the adaptability of the species in an ecosystem to prolong the cycles of boom and bust. The mied-mode oscillations in our model arise due to the combination of the generalied canard mechanism along with a singular Hopf bifurcation in the system. AMS Subject Classifications. 70K70 34D5 37G0 92D25 92D40. INTRODUCTION Ecological models with multiple time scales have been studied over the last few decades and have received considerable interest in the ecological community. Forest pest models involving slow dynamics of trees and fast dynamics of pests Brøns & Kaasen 200 tritrophic food chain models with time diversification Deng 200 age-structured predatorprey models with dormancy of the predators Kuwamura & Chiba 2009 are some such eamples. Two predators competing for a common prey with the assumption that the prey ehibits fast dynamics naturally involves two time-scales and was first considered by Muratori & Rinaldi Muratori & Rinaldi 989. They proved that the three species could coeist by the mechanism of cyclic behavior i.e. under high eploitation pressure the prey population collapses and remains at an endemic level until the predators become scarce. The prey then quickly regenerate and the abundance of prey provide a chance to the predators to grow slowly. The cycle repeats when the eploitation pressure reaches a threshold. However in reality species can coeist ehibiting small fluctuations in their densities without eperiencing strict periodic outbreaks and collapses or perhaps adapt themselves in a way to prolong the cycles of boom and bust. In this paper we consider a two-trophic model involving two time-scales that addresses both the features discussed above. The first feature is represented by an attracting small amplitude limit cycle that arises due to a singular Hopf bifurcation in the system. The second feature is manifested through mied-mode oscillations MMOs which are concatenations of large amplitude oscillations LAOs and small amplitude oscillations SAOs. In models with multiple time-scales such as in chemical biological and neuronal systems comple dynamics such as chaos and mied-mode oscillations have been commonly observed Brøns et. al and the references therein. However in ecological model this behavior has been very rarely observed. The model that we consider reads as under the initial conditions dx dτ dy dτ dz dτ = rx X K dy = b p XY H +X = b 2p 2 XZ p XY H +X p 2XZ H 2 +X H 2 +X d2z sz2 X0 = X 0 0 Y 0 = Y 0 0 Z0 = Z where X represents the population density of the common prey and Y Z represent the densities of the predators. The density dependent mortality term sz 2 of the second predator describes the presence of an intraspecific competition commonly attributed to territorial species. The parameters r and K represent the intrinsic growth rate and the carrying capacity of the prey p is the maimum per-capita predation rate of Y H is the semi-saturation density so that the predator s predation rate is half of its maimum p /2 when X = H b is the birth-to-consumption ratio of Y and d is the per-capita natural death rate of Y. The other parameters p 2 b 2 d 2 H 2 are defined analogously. Ruan et al. Ruan et al studied model and showed that two competitors ehibiting Holling II functional response with a density dependent mortality in one of the predators can coeist upon the same prey. In the absence of Dynamic Publishers Inc.

2 22 S. Sadhu density dependent mortality Hsu Hubbell and Waltman Hsu et al. 978 also see Muratori & Rinaldi 989 showed that two competitors can coeist only via a locally attracting periodic orbit. Hence if s = 0 in system then there eists no positive equilibrium state and thus the system is only weakly persistent Hutson & Schmitt 992. However when s > 0 Ruan et al. in Ruan et al showed the eistence of a positive steady state which is also globally attracting thus ruling out the principle of competitive eclusion Hardin 960; MacArthur & Levins 964. With the following change of variables and parameters: t = rτ = X K y = py rk = p2z bp ζ = rk r ζ2 = b2p2 H β = r K d 2 d = h = srk b 2p 2 b 2p 2 2 system takes the following dimensionless form: ẋ = y β + β 2 + ẏ = ζ y c β + ż = ζ 2 d h β 2 + β2 = H2 K c = d b p where = d. Similar scaling variables were first considered in a three-trophic food chain model by Deng Deng dt 200. We assume that the maimum per-capita growth rate of the prey is much larger than that of the predators namely r >> ma{b p b 2p 2} or equivalently 0 < ζ ζ 2 <<. For simplicity we consider that ζ = ζ 2 = ζ. The parameters c and d measure the ratios of the death-rate to the birth-rate of Y and Z respectively and β β 2 measure the predating efficiencies of the two predators. We assume that 0 < c d < otherwise the predators would die out faster than they could reproduce even at their maimum reproduction rate and 0 < β β 2 < which means that the predators will reach half of their maimum predation rates before the prey reaches its carrying capacity. The parameter h measures the strength of the intraspecific competition in the class of Z. Thus we obtain a singular perturbed system in two-time scales with being the fast variable and y and being the slow variables. 2 THE GEOMETRIC SINGULAR PERTURBATION APPROACH In this section we perform a geometric analysis of system 3. On rescaling t ζt 3 transforms to ζẋ = y β + β 2 + ẏ = y c β + ż = d h β 2. + System 3 is referred to as the fast-system and 4 as the slow system. As ζ 0 the trajectories of 3 during fast epochs approach to the solutions of the layer equations given by 5. On the other hand during slow epochs trajectories of 4 converge to the solutions of the reduced problem given by 6. ẋ = y ẏ = 0 ż = 0. β + β = y β + ẏ = y c β + ż = d h β 2 + β 2 + The fast and slow subsystems 5 and 6 can be used to understand and study the dynamics of the full system 3 or 4. The algebraic equation in 6 defines the critical manifold M = { y : = 0 or u y = 0} := T S where T = {0 y : y 0} S = { y : u y = 0} and u y = y/β + /β 2 + is the nontrivial -nullcline. The points of M are equilibrium points of the layer equations 5. The critical manifold shown in Figure is a 2-dimensional surface in R 3 consisting of two normally attracting sheets and two normally repelling sheets. The surface S and the plane T intersects along the fold line F = { y : y/β + /β 2 = = 0}. The surface S can be written as S a S r F where S a = { y S : u < 0} is the attracting sheet S r = { y S : u > 0} is the repelling sheet and F = { y S : u = 0 u 0} is the fold curve. If β β 2 we have F = { β + 2 β β 2 β 2 2 β 2+ 2 β β β hand when β = β 2 then F = : ma{β β 2 } { y S : = β y + = +β min{β β 2 } }. On the other }. It follows from geometric singular perturbation theory Fenichel 979 that normally hyperbolic segments of M perturb smoothly to nearby segments of the slow manifold M ζ. The segments corresponding to S a and S r will be denoted by Sζ a and Sζ r respectively. Since u 0 by the implicit function theorem the surface S can be locally written as a graph of = φ y i.e. u y φ y = 0. Differentiating u y = 0 implicitly with respect to time gives us the relationship u ẋ+u yẏ + u ż = 0. Denoting the nontrivial y and -nullclines by v and w respectively the reduced flow 6 restricted to S reads as uẋ ẏ uyyv + u w = yv. 7 =φy

3 Canards and Mied-Mode Oscillations 23 System 7 has singularities along the fold curve F. Hence standard eistence and uniqueness results do not hold. Different solutions can approach the same point on F in finite time. Note that the reduced flow is directed towards the fold or away from it. On rescaling the time by the factor u 7 transforms to the desingularied system ẋ = ẏ uyyv + u w u yv. 8 =φy Points on F for which u yyv +u w 0 are called jump points Popović At these points solutions eit into relaation after reaching F giving rise to relaation dynamics see Figure 2. On the other hand points of F that satisfy u yyv + u w = 0 are known as folded equilibria of of 8. They can be classified as folded nodes or folded saddles or folded saddle-nodes depending on the signs of the eigenvalues of the Jacobian evaluated at these points. At these points the reduced flow can cross from S a ζ to S r ζ which is a prerequisite for the eistence of canards see Figure 3. Canards are trajectories that flow along the attracting slow manifold passes close to a non-hyperbolic point of the critical manifold and then spend an O amount of time on the repelling sheet of the slow manifold Smolyan & Wechselberger 200. T a =0 S r S a uy=0 Fold curve Fast flow Slow flow T r Figure : The critical manifold M with its attracting sheets represented by S a and T a and the repelling sheets by S r and T r. Figure 2: A typical relaation oscillation. Singular Funnel S r S a S r S a a Desingularied flow corresponding to 8 b A schematic view of the reduced flow corresponding to 7 Figure 3: Eistence of a folded node singularity black dot of 8 for β = 0.5 β 2 = 0.25 c = 0.4 d = 0.25 h =.25. Here γ s and γ w represent the strong and the weak eigendirections respectively of the lineariation of 8 at the folded node. 3 ANALYSIS OF THE SUBSYSTEMS 3. Analysis of the y-subsystem In the absence of the territorial predator we obtain the subsystem: ζẋ = y ẏ = y β + β + c. 9 We note that 9 is the singularly perturbed Rosenweig-MacArthur model Rosenweig & MacArthur 963 whose dynamics are well-known. We omit the details here.

4 24 S. Sadhu 3.2 Analysis of the -subsystem In absence of the predator y we obtain the subsystem which is a singularly perturbed Baykin s model Baykin 2000: ζẋ = β 2. + ż = d h 0 β 2. + The subsystem 0 allows the eistence of multiple equilibria and multiple limit cycles Hopf homoclinc and Bogdanov- Takens bifurcations. A two dimensional bifurcation diagram in the d-β 2 parameter space with some of the corresponding trajectories are shown in Figure 4. For details see Kunetsov 998 & Baykin a b 3 A 2a a d BT 2b P b 2a 2b 2c 3 2c CP GH β 2 a Parametric Portrait of 0 in the d-β 2 space. b Phase portraits of 0. Figure 4: A 2-parameter bifurcation diagram with the corresponding phase portraits of the subsystem 0 at h =.25 and ζ = 0.0. In a: the solid black curve represents the Hopf curve the dashed curve represents the cusp bifurcation of the equilibrium points and the dotted curve represents the saddle-node bifurcation of limit cycles. Notations used: BT - Bogdanov-Takens bifurcation CP - cusp bifurcation GH - generalied-hopf bifurcation the first Lyapunov coefficient is ero A - big homoclinic loop P - homoclinic orbit at the saddle fied point. In b: the subsystem has eactly one equilibrium state which is a stable node in region a. In b the equilibrium state converts to a spiral node and two limit cycles are born. In 3 the equilibrium state loses its stability and an attracting stable limit cycle governs the dynamics. In 2a three equilibria states eist: one stable one saddle and one unstable. In 2b an unstable limit cycle is born around the stable equilibrium state along with a large stable limit cycle. In 2c the stable equilibrium state loses its stability through a subcritical Hopf bifurcation. The large limit cycle persists and is globally attracting. Following the approach of Baer & Erneu 986; Braaksma 998 we identify that the subsystem 0 admits a singular Hopf bifurcation. To verify that we rescale t t/β + in 0 and obtain the following orbitally equivalent system: ζẋ = β 2 + := u ż = d dβ 2 hβ 2 + := w. Suppose that be such that u = w = 0. With h as the varying parameter assume that h is the parameter value at which u h = 0 for fied values of d and β 2. Then we obtain that = β2 = 2 + β22 4 h 4 d β2 dβ2 =. + β 2 3 We will verify that system satisfies all the conditions of Theorem of [3] Braaksma 998. Clearly u = 0 w = 0 and u = 0. Moreover 2 u 2 = β 2 0 u h = β2 0 2 h 2 w β2 + β2 = 0 w h 2 h = h 32 + β u u det w w h = 4 β2 β2 2 0.

5 Canards and Mied-Mode Oscillations 25 Finally [ u u u u w w = 6β 2 β 2 + β u h w h + u h ] h Applying Theorem 2 of Braaksma 998 we conclude that system undergoes a Hopf bifurcation Oζ away from h in the parameter space and Oζ away from in the phase space. Moreover the eigenvalues have singular imaginary parts since u w h < 0 for details see Proposition of [3]. Moreover the first Lyapunov coefficient l can be calculated using the formulas in Baer & Erneu 986 which in this case is l = d β2 dβ2 2 β 2 β 2 2 β 2 + β 2 β 2 2 β2 β β 2 The sign of l determines the direction of bifurcation: the bifurcation is subcritical if l > 0 and supercritical if l < Canard Eplosion Subsystem 0 also ehibits a canard eplosion. At the Hopf bifurcation a periodic orbit is born with an intermediate period between the fast Oζ and slow O time scales. The sie of this periodic orbit grows rapidly from diameter Oζ to diameter O in an eponentially small parameter region. Canard eplosion refers to this eponential increase in the sie of the limit cycles over an etremely narrow interval Figure 5. The LAOs are known as relaation oscillations. During these cycles due to heavy predatory eploitation the prey population collapses and remains in an endemic state until the predators become scarce after which the prey quickly regenerate allowing the predators to grow slowly. Since the critical manifold S is S-shaped and conditions 2-3 hold on applying the results of Desroches et al 202 Theorem 2.2 we conclude that a canard eplosion occurs at Oζ from the Hopf point in the subsystem 0. a Amplitude vs h b A small periodic orbit at h = c A large periodic orbit at h = Figure 5: Amplitude of vs h representing the canard eplosion of the subsystem 0 with β 2 = 0.25 d = 0.25 and ζ = 0.0. The system undergoes a supercritical Hopf bifurcation at h A small periodic orbit is born which grows to a large relaation cycle over an eponentially narrow parameter interval. 4 ANALYSIS OF THE FULL SYSTEM We will analye the full-system 4 in this section. We note that 4 admits an interior equilibrium point E = y where = [ ] cβ c c cβ dβ2 c cβ d y = β + c 2 hcβ + cβ provided that = hold. Also we assume that cβ d dβ2 c 5 hcβ + cβ 2 cβ d dβ 2 c > 0 and c cβ cβ d dβ2 c >. 6 c 2 hcβ + cβ 2 2 Treating β β 2 c d as fied parameters we define a constant h by β + c c > 0. 7 h = c3 β β 2cβ d dβ 2 c. 8 cβ + cβ 2 3 cβ + β + c

6 26 S. Sadhu Proposition. Keeping β β 2 c d fied under the assumptions 6-7 the interior equilibrium point E is locally asymptotically stable if h h for all ζ > 0 and unstable if h < h provided ζ > 0 is sufficiently small. Proof. The proof follows by performing a linear stability analysis of 4 at E. Indeed the characteristic equation of the Jacobian evalauted at E reads as where Q = u y ζ + h Q 2 = ζ λ 3 + Q λ 2 + Q 2λ + Q 3 = 0 h u y + β y β + + β2 Q 3 β = βh y ζβ +. 3 Applying the Routh-Hurwit criterion it follows that E is locally asymptotically stable if u y 0 for all ζ > 0 and unstable if u y > 0 provided ζ > 0 is sufficiently small. With some calculations we note that u y β + c c h = β h and thus the result follows under assumption 7. The detailed calculations can be found in Sadhu in progress. The equilibrium state E loses its stability at h = h + Oζ for sufficiently small ζ. In fact the system undergoes a singular Hopf bifurcation at Oζ away from h in the parameter space and Oζ away from the fold curve in the phase space. Theorem. Under the assumption that β β 2 c d are fied and conditions 6-7 hold system 4 undergoes a singular Hopf bifurcation at h = h + Oζ at a distance Oζ away from the fold curve F. The direction of the Hopf bifurcation is determined by the sign of the first Lyapunov coefficient l l > 0 corresponds to the subcritical bifurcation and l < 0 corresponds to the supercritical bifurcation: 2 y β l = Fuuu FuwH w H uu 9 2 where F uuu = 24ω 2 2 y β βy 3 β β + + β 2 + β2h 2 4 β ω 2 β β β + 2 y + β2 β2 + 2 β + 4 β F uw = H w = 2 β ω 2 y β β β 2 ω 2 β h β 2 + β β 2 y H uu = ω 2 β + 4 β y and + ω 2 = β 2 ω 2 β β y β + + β2 3 β β β 2 β β β β 2 β β β 2 + y β h β 2 + h ω 2 β Proof. The proof follows from the work of Braaksma 998. With some tedious work it can be shown that 4 satisfies all the conditions of Theorem and Theorem 2 of [3] Braaksma 998 thus allowing the eistence of a singular Hopf bifurcation. The first Lyapunov coefficient l can be calculated using the formula 5 in Braaksma [3] Braaksma 998. The details of the proof can be found in the upcoming paper Sadhu in progress. Suppose that the parameters β β 2 c d and h are so chosen that 6-7 holds and that l < 0 where l is given by 9. Then at the Hopf point a stable periodic orbit is born as shown in Figure 6. The periodic orbit can either grow into a relaation cycle see Figure 6 a or may lose its stability and undergo a series of period doubling bifurcations and stabilie into an orbit which is a concatenation of SAOs and LAOs known as a mied-mode oscillation MMO see Figure 6 b and Figure 9. MMOs are denoted by signatures {L k i i } i where each pair L k i i denotes a segment of the MMO comprised of L i large oscillations followed by k i small oscillations. SAOs can be viewed as a 0 MMO pattern while LAOs can be viewed as a 0 MMO pattern. Several distinct mechanisms from dynamical systems and bifurcation theory have been used to eplain the mechanism of MMOs Desroches et al. 202 and the references therein.

7 Canards and Mied-Mode Oscillations 27 Interior Equilibrium SN Equilibrium of the subsystem HB Interior Equilibrium PD PD Equilibrium of the subsystem HB SN h a d = 0.3. PD PD h b d = 0.26 Figure 6: A bifurcation diagram for the system 4: over the bifurcation parameter h. In both the diagrams we have chosen ζ = 0.0 β = 0.5 β 2 = 0.25 and c = Notations used: SN - saddle-node bifurcation of limit cycles HB - Hopf bifurcation PD- period-doubling bifurcation. The green blue circles represent the stable unstable periodic orbits. In both cases the Hopf bifurcation is supercritical giving birth to a stable periodic orbit. In a the stable periodic orbit grows in sie as h decreases and gradually converts to a relaation oscillation. In b as the stable periodic orbit grows in sie and converts to a relaation oscillation it loses its stability in between. The parameter regime where it becomes unstable mied-mode oscillations are born. The figures were generated through XPPAUT Ermentrout h Figure 7: A detailed -parameter bifurcation diagram in MATLAB for system 4 where the onset of MMOs is first observed. The black dots red represent the local maima minima of. Here d = 0.26 and the other parameter values are the same as in Figure 6. A Hopf bifurcation occurs at h = where a small periodic orbit 0 is born. The 0 orbit goes through two period-doubling bifurcations and stabilies into a MMO pattern at around h =.75. The periodic orbit loses its stability through a series of period-doubling bifurcations and finally stabilies to a relaation oscillation 0. Figure 8: A detailed bifurcation diagram for system 4: over the bifurcation parameter h. Here d = 0.2 and the other parameter values are the same as in Figure 6. The 0 periodic orbit goes through a period-doubling route to chaos and then stabilies into a 2 MMO pattern which then later loses stability and stabilies into a MMO pattern. The periodic orbit undergoes a series of period-doubling and saddle-node bifurcations and stabilies into a 2 MMO pattern which finally transforms to a relaation oscillation. In systems with one fast and two slow variables where the critical manifold is S-shaped MMOs can occur when the desingualried system 8 admits a folded node equilibrium in addition to the eistence of a singular periodic orbit which consists of fast fibers of the layer problem a global return segment and a segment on S a with one end end at the folded node that lies in the interior of a singular funnel Brøns et al If a trajectory gets trapped in a singular funnel see Figure 3b which is formed by the fold curve and the strong eigendirection near the folded node singularity then it eperiences some delay caused due to the rotational properties of the weak canard until it jumps to the other attracting

8 28 S. Sadhu a An MMO periodic orbit with signature 2. b The corresponding time series in Figure 9: Here ζ = 0.0 β = 0.5 β 2 = 0.25 c = 0.38 d = 0.2 h = sheet of the critical manifold Sadhu 205. A global return mechanism then brings the orbit back to the singular funnel. MMOs can also arise due to the combination of a singular Hopf bifurcation and a weak return mechanism Guckenheimer The unstable manifold of the equilibrium E which is a saddle-focus after the bifurcation can interfere with the slow dynamics on S r ε giving rise to a small amplitude oscillation. We find that the MMOs in system 4 occur due to the combination of a singular Hopf bifurcation and the generalied canard phenomenon. More detail studies regarding the nature of the MMOs the corresponding Farey sequence and chaotic dynamics are being pursued Sadhu & Chakraborty Thakur 205; Sadhu in progress. 5 ACKNOWLEDEMENT The author would like to thank Dr. S. C. Thakur of University of California at San Diego for carefully reading the manuscript helping with some of the numerical figures and for many insightful discussions. REFERENCES [] S.M. Baer and T. Erneu Singular Hopf bifurcation to relaation oscillations SIAM J. Appld. Math [2] A.D. Baykin Nonlinear Dynamics of Interacting Populations World Scientific Singapore 998. [3] B. Braaksma Singular Hopf Bifurcation in Systems with Fast and Slow Variables J. Nonlinear Sci [4] M. Brøns and R. Kaasen Canards and mied-mode oscillations in a forest pest model Theoretical Population Biology [5] M. Brøns T. J. Kaper and H. G. Rotstein Introduction to Focus Issue: Mied Mode Oscillations: Eperiment Computation and Analysis Chaos [6] B.M. Brøns M. Krupa and M. Wechselberger Mied Mode Oscillations Due to the Generalied Canard Phenomenon Fields Institute Communications [7] B. Deng Food chain chaos due to junction-fold point Chaos [8] M. Desroches J. Guckenheimer B. Krauskopf C. Kuehn H. M. Osinga and M. Wechselberger Mied-Mode Oscillations with Multiple Time Scales SIAM Review [9] B. Ermentrout Simulating Analying and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students SIAM [0] N. Fenichel Geometric singular perturbation theory for ordinary differential equations Journal of Differential Equations [] J. Guckenheimer Singular Hopf Bifurcation in Systems with Two Slow Variables SIAM Journal of Applied Dynamical Systems [2] G. Hardin Competitive eclusion principle Science [3] S.-B. Hsu S.P. Hubbell and P. Waltman Competing predators SIAM J. Appl. Math [4] V. Hutson and K. Schmitt Permanence and the dynamics of biological systems Math. Biosci [5] M. Kuwamura and H. Chiba Mied-mode oscillations and chaos in a prey-predator system with dormancy of predators Chaos [6] Y.A. Kunetsov Elements of Applied Bifurcation Theory Springer 998.

9 Canards and Mied-Mode Oscillations 29 [7] R. MacArthur and R. Levins Competition habitat selection and character displacement in a patchy environment Proc. Natl Acad. Sci. USA [8] S. Muratori and S. Rinaldi Remarks on competitive coeistence SIAM J. Applied Math [9] N. Popović Mied-mode dynamics and the canard phenomenon: towards a classification Journal of Physics: Conference Series [20] M.L. Rosenweig and R. MacArthur Graphical representation and stability conditions of predator-prey interactions American Naturalist [2] S. Ruan A. Ardito b P. Ricciardi and D.L. DeAngelis Coeistence in competition models with density-dependent mortality Comptes Rendus Biologies [22] S. Sadhu Mied mode oscillations and chaotic dynamics in a two-trophic ecological model with Holling Type II functional response in press Bulletin of Calcutta Mathematical Society 205. [23] S. Sadhu Analysis of mied-mode oscillations in a singularly perturbed two-trophic ecological model in progress. [24] S. Sadhu S. Chakraborty Thakur Mied-Mode Oscillations and Chaotic Dynamics in a Predator-Prey-Scavenger Ecosystem Proceedings of IMBIC [25] P. Smolyan and M. Wechselberger Canards in R 3 Journal of Differential equations