Refuge use as a function of predator-prey encounters Betsabé González-Yañez1 , Eduardo González-Olivares1 and Rodrigo Ramos-Jiliberto2 1
Grupo Ecología Matemática, Instituto de Matemáticas, Ponti…cia Universidad Católica de Valparaíso. Casilla 4950, Valparaíso, Chile
[email protected],
[email protected], http://ima.ucv.cl
2
Departamento de Ciencias Ecológicas, Facultad de Ciencias, Universidad de Chile Las Palmeras 3425 Ñuñoa. Casilla 653, Santiago, Chile
[email protected],
August 19th, 2005 Abstract In earlier works it has been considered that the use of refuges by a fraction of prey exerts a stabilizing e¤ect in the dynamics of the interacting populations. In this article, we test the above statement assuming that the quantity of prey in refugia is poportional to encounters between prey and predators and we analyze the dynamic properties of such a system through modifying the well-known Lotka-Volterra predator-prey model with prey self-limitation. We obtain the existence of a parameter set in which the system presents three positive equilibrium points, two of which are attractors, that is, depending on the initial conditions the trajectories can have diferent !limits. Biologically this phenomenon allows for a number of natural interesting behaviors, such as outbreaks. However, we demonstrate that for most parameter values, there exists an unique and positive equilibrium point which is globally asymptotically stable.
Key Word: predator prey model, refuge, stability, bifurcation. AMS Classi…cation 92D25, 34C, 58F14, 58F21
1
1
INTRODUCTION
One of the more relevant behavioral traits that a¤ect the dynamics of predatorprey systems is the use of spatial refuges by the prey. Spatial refuges are found where environmental heterogeneity provides less-accessible sites for predators, which can be exploited or utilized by a given number of prey. In this way, some fraction of the prey population is partially protected against predators [12]. According to Taylor [27] the di¤erent kinds of refuges can be arranged into three types: a) those which provides permanent spatial protection for a small subset of the prey population, b) those which provide temporary spatial protection, and c) those which provide a temporal refuge in numbers, i. e. decrease the risk of predation by increasing the abundance of vulnerable prey. Within population ecologists, it has been common to a¢ rm that the use of refuges by a fraction of the prey population exerts a stabilizing e¤ect when a deterministic continuous time model is employed for describing the predatorprey interaction [19, 23, 24]. This a¢ rmation is veri…able by considering two types of refugia in a well kwown Lotka-Volterra model: one for which the quantity of prey in cover xr is proportional to population size x = x(t), that is,xr = x; or else, one for which the refuged population is a …xed quantity xr = [19, 14, 27]. These results can also be extended when the prey population exhibits a selfregulated growth, i.e. the prey growth is described by the logistic equation [8] (Lotka-Volterra model with self-regulation). In such cases there exists a unique equilibrium point at the interior of the …rst quadrant which is globally asymptotically stable [6, 11, 12]. This property can be proved constructing a Lyapunov function [2, 9, 10, 16, 17]. The general conclusions from these studies is that refugia protecting a constant number of prey lead to a stronger stablizing e¤ect on population dynamics, compared to refugia protecting a constant proportion of prey [26]. In [6, 12] it is shown that the simplistic interpretation of the stabilizing role of refugia may not be correct in general, since for more complex models the use of refuges can exert a locally destabilizing e¤ect, due to emergence of stable limit cycles or an oscillatory behavior of populations [26]. Other model considering prey refuges can be found in [20, 23]. In particular, Ruxton derives a continuous time model under the assumption that the rate at which prey move to the refuge is proportional to predator density, that is xr = y, showing that this behavior has a stabilizing e¤ect. On the other hand, McNair presents a study of systems with more complex interactions incorporating prey refuges which are analyzed in [15], who proves the existence of an unique limit cycle. In the present work, we analyze the population consequences of refuge use in a Lotka-Volterra model with self-limitation, assuming that the quantity of prey in refugia is proportional to encounters between prey and predators, that is xr = xy, where x = x(t) and y = y(t) indicate predator and prey population size 2
for t 0. This approach also allows the use of more sophisticated mathematical functional forms, such as saturating monotonic functions.
2
THE MODEL
The oldest predator-prey model that assume the e¤ects of refuge for a part of prey population is proposed in [19], making a modi…cation to well-known Lotka-Volterra predator-prey model concluding that the refuge has a stabilizing e¤ect in the interaction because the neutral existent equilibrium (ciclic trajectories surrounding an unique equilibrium point) is transformed in a globally asymptotically stable equilibrium point. Later in [19, 23] it is assumed that the prey have a logistic growth and it is proposed the interaction is determined by the quantity of disposable prey x xr . Considering also that populations are uniformly distributed in the environment, abiotic phenomenon not in‡uence on the growth of both populations, and nor sex and age structure are assumed, the interaction is described by a continuous time model given by the autonomous di¤erential equations X :
dx dt dy dt
= r (1 = (p (x
x K)
x q (x xr ) y xr ) c)y
(1)
with 0 xr x and = (r; K; q; p; c) 2 0 or C < 1 2E and we have 2 ((1 E 2C) (E C)) (1 E) (E C) 3EC C E 2 + E > 0 2 4C (1 C) (E C) > 0. which is always true for C < E < 1. 1 E a2.2) F < 0, if and only if, 1 E 2C 0 or C 2 but in this case we obtain that 1 E D = 3EC C E 2 + E = 3E 1 2E E2 + E 2 1 = 2 (1 E) (E + 1) < 0 therefore u2 is not a real numberl. Then, u2 > C. a3) Considering the diference u3 1, that is, p u3 1 = 2(E1 C) W (1 + E) (E C) a3.1) Assuming that this diference is negative; it has now that p G = W (1 + E) (E C) < 0 or, 2 (1 E) 3EC C E 2 + E (1 + E) (E C) < 0 4E 2 (1 C) < 0 which is always true for C < E < 1 a3.2) Assuming that this diference is positive the contradiction 4E 2 (1 C) > 0: is obtained, then u3 < 1 >From a1, a2 and a3 we conclude that u1 , u2 and u3 2 ]C; 1[, and the three equilibrium point is at region . b. Relation between u1 and u3 . Considering the diference p u1 u3 = 2(E1 C) (3E 1) (E C) W after a tediousus algebra we can see that it can positive or negative, then the position between the respective absise of this equilibrium points is u1 < u3 or u1 > u3 , depending of quantity r = 2E 2 3EC + 2C E, a new bifurcation curve. c. Relation between u1 and u2 Considering the diference 9
p u1 u2 = 2(E1 C) (3E 1) (E C) + W we newly obtain that the relative position dependent of r. We de…ne n the subregions of parameter space o 1 2E 1 E 1 = (E; C) 2 = E < C < E and E < 1 2 3E 3E+1 3 n o 1 2E 1 E = E 2 3E < C < E 3E+1 and E > 31 2 = (E; C) 2 where n o 2 1 E = (E; C) 2 0, or, C < E 2E ) 3E 2 , the point (E; C is local atractor and it has the order u1 < u2 < u3 o u2 < u3 < u1 . b) If r = 2E 2 3EC + 2C E < 0, the point (E; (1 CE)E ) is saddle point and it has the order u2 < u1 < u3 . Proof: >From theorem 6 it has that detDZ N (E; (1 CE)E ) = B 2E 2 3EC + 2C E then, the behavior of this equilibiurm point is dependent of quantity r = 2E 2 3EC + 2C E. Theorem 9 Nature of equilibrium points (u2 ; v2 ) and (u3 ; v3 ). a) Suppose that r = 2E 2 3EC +2C E > 0, it has the order u1 < u2 < u3 or u2 < u3 < u1 . a1) If it has the order u1 < u2 < u3 , the point (u2 ; v2 ) is saddle point and (u3 ; v3 ) is local atractor. a2) If it has the order u2 < u3 < u1 , the point (u3 ; v3 ) is saddle point and (u2 ; v2 ) is local atractor. b) If r = 2E 2 3EC + 2C E < 0, the points (u2 ; v2 ) and (u3 ; v3 ) are local atractor. Proof Considering theorem 8 about the nature of equilibrium point (E; (1 CE)E ) and because trDZ N (u; v) = u B (u C) < 0.
4
CONCLUSIONS
The model analyzed here is a form of the Lotka-Volterra model with selflimitation, incorporating the use of refuge by a fraction of prey and assuming that this quantity is proportional to population sizes of both species. The latter assumption seems reasonable, since constitutes a good and tractable approximation to observed behaviors related to inducible defenses [22]. On the other hand, in [23] it is assumed that the quantity of prey searching for refuges is proportional to the abundancwe of predators. Mathematically, the studied system belongs to a Kolgomorov type [8], since the coordinate axes are invariant sets and the model is a Gause type [8] conforming the mass-action principle [4]. We verify that for any parameter value the equilibrium points (0; 0) and (K; 0) are saddle points, and this imply that the populations always coexist. Moreover, there exists a wide range of parameter values for which the system has an unique equilibrium point (E; ve ) being globally asymptotically stable. Our results are in agreement with the current ecological theory, in that refuge use exert a stabilizing e¤ect in predator-prey interactions. Our results also reveal the existence of a region within the parameter space for which three positive equilibrium points coexists. In this case, two equilibrium points are attractors (locally asymptotically stable points) and the third one is
11
a saddle point. A separatrix curve appears in the phase plane, determined by the stable manifold of the saddle point, that split the state-variable space into two basins of attraction. The existence of two attracting equilibrium points is an interesting property that could be understood as a possible endogenous source of outbreaks which exhibits some irruptive populations, and a possible source of non-stationary dynamics in stochastic environments. ACKNOWLEDGMENTS: This work was supported by grants FONDECYT 1040833 and DI UCV No 124 794/2004 to EGO and FONDECYT 1040821to RRJ. The authors wish to thank the members of the Mathematical Ecology Group at the Instituto de Matemáticas of the Ponti…cia Universidad Católica de Valparaíso, for their valuable comments and suggestions.
References [1] A. A. Andronov, E. A. Leontovich, I. Gordon and A. G. Maier, Qualitative theory of second-order dynamic systems. A Halsted Press Book, John Wiley and Sons, New York, 1973. [2] A. Ardito and P. Ricciardi, Lyapunov functions for a generalized Gausetype model. Journal of Mathematical Biology Vol 33, 816-828, 1995. [3] D. K. Arrowsmith and C. M. Place, Dynamical Systems: Di¤erential equations, maps and chaotic behaviour. Chapman and Hall, 1992. [4] A. A. Berryman, A. P. Gutierrez, and R. Arditi. Credible, parsimonious and useful predator-prey models - A reply to Abrams, Gleeson, and Sarnelle. Ecology 76 (6), 1980-1985, 1995 [5] C. Chicone, Ordinary di¤erential equations with applications, Texts in Applied Mathematics 34, Springer, 1999. [6] J. B. Collings, Bifurcations and stability analysis of a temperaturedependent mite predator-prey interaction model incorporating a prey refuge, Bulletin of Mathematical Biology 57, 63-76, 1995. [7] F. Dumortier, Singularities of vector …elds. Monografías de Matemática No 32, Instituto de Matemática Pura e Aplicada IMPA, Brazil, 1978. [8] H. I. Freedman, Deterministic Mathematical Model in Population Ecology. Marcel Dekker, .1980. [9] T. C. Gard, A new Liapunov function for the simple chemostat, Nonlinear Analysis, RWA, Vol. 3, 211-226, 2002. [10] B-S. Goh, Management and Analysis of Biological Populations. Elsevier Scienti…c Publishing Company, 1980.
12
[11] E. González-Olivares and R. Ramos-Jiliberto, Dynamic consequences of prey refuges in a simple model system: more prey, fewer predators and enhanced stability, Ecological Modelling, Vol. 166, Issues 1-2, 135-146, 2003. [12] E. González-Olivares and R. Ramos-Jiliberto, Consequences of prey refuge use on the dynamics of some simple predator-prey models: Enhancing stability? Proceedings of the Third Brazilian Symposium on Mathematical and Computational Biology, In R. Mondaini (ed) E-Papers Serviços Editoriais, Ltda., Rio de Janeiro, Vol. 2, 75-98, 2004. [13] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag, 1983. [14] G. W. Harrison, Global stability of predator-prey interactions. Journal of Mathematical Biology 8, 139-171, 1979. [15] A. R. Hausrath, Analysis of a model predator-system with refuges, Journal of Mathematical Analysis and Applications 181, 531-545, 1994. [16] A. Korobeinikov, A Lyapunov function for Leslie-Gower predator-prey models, Applied Mathematical Letters 14, 697-699, 2001. [17] T. Lindström, Global stability of a model for competing predators: An extension of the Ardito and Ricciardi Lyapunov function, Nonlinear Analysis, Vol. 39, 793-805, 2000. [18] R. M. May, Stability and complexity in model ecosystems. Princeton University Press, 1974. [19] J. Maynard Smith, Models in Ecology. Cambridge, At the University Press, 1974. [20] J. M. McNair, The e¤ects of refuges on predator-prey interactions: a reconsideration. Theoretical population biology, 29, 38-63, 1986. [21] N. Minorsky, 1962, Nonlinear oscillations, Van Nostrand Company Inc. [22] R. Ramos-Jiliberto and E. González-Olivares, Relating behavior to population dynamics: a predator-prey metaphysiological model emphasizing zooplankton diel vertical migration as an inducible response. Ecological Modeling 127, 221-233, 2000. [23] G. D. Ruxton, Short term refuge use and stability of predator-prey models. Theoretical population biology, 47, 1-17, 1995. [24] A. Sih, Prey refuges and predator-prey stability. Theoretical Population biology, 31, 1-12, 1987. [25] J. Sotomayor, Lições de Equações Diferenciais Ordinárias. Projeto Euclides IMPA, CNPq., 1979.
13
[26] P. D. N. Srinivasu and I. L. Gayitri, In‡uence of prey reserve capacity on predator-prey dynamics, Ecological Modeling 181, 191-202, 2005. [27] R. J. Taylor, Predation. Chapman and Hall, 1984. [28] P. Turchin, Complex population dynamics: a theoretical/ empirical synthesis, Princeton University Press, 2003.
14
