Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator-prey model. (English) Zbl 1519.92185
Summary: Time delay is an essential ingredient of spatio-temporal predator-prey models since the reproduction of the predator population after predating the prey will not be instantaneous, but is mediated by a constant time lag accounting for the gestation of predators. In this paper we study a predator-prey reaction-diffusion system with time delay, where a stability analysis involving Hopf bifurcations with respect to the delay parameter and simulations produced by a new numerical method reveal how this delay affects the formation of spatial patterns in the distribution of the species. In particular, it turns out that when the carrying capacity of the prey is large and whenever the delay exceeds a critical value, the reaction-diffusion system admits a limit cycle due to the Hopf bifurcation. This limit cycle induces the spatio-temporal pattern. The proposed discretization consists of a finite volume element (FVE) method combined with a Runge-Kutta scheme.
MSC:
| 92D25 | Population dynamics (general) |
| 35K57 | Reaction-diffusion equations |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
| 35B35 | Stability in context of PDEs |
Keywords:
spatio-temporal patterns; time delay; limit cycle; pattern selection; finite volume element discretizationSoftware:
PRED_PREYReferences:
| [1] | Andreianov, B.; Bendahmane, M.; Ruiz-Baier, R., Analysis of a finite volume method for a cross-diffusion model in population dynamics, Math. Models Methods Appl. Sci., 21, 02, 307-344 (2011) · Zbl 1228.65178 |
| [2] | Bellen, A.; Zennaro, M., Numerical Methods for Delay Differential Equations (2003), Clarendon Press, Oxford University Press: Clarendon Press, Oxford University Press New York · Zbl 1038.65058 |
| [3] | Bencheva, G., Comparative analysis of solution methods for delay differential equations in hematology, (Lirkov, I.; etal., LSSC 2009. LSSC 2009, Lect. Notes Comput. Sci., vol. 5910 (2010), Springer-Verlag: Springer-Verlag New York), 711-718 · Zbl 1280.34052 |
| [4] | Bendahmane, M.; Anaya, V.; Sepúlveda, M., Mathematical and numerical analysis for predator-prey system in a polluted environment, Netw. Heterog. Media, 5, 4, 813-847 (2010) · Zbl 1262.35123 |
| [5] | Bendahmane, M.; Ruiz-Baier, R.; Tian, C., Turing pattern dynamics and adaptive discretization of a superdiffusive Lotka-Volterra system, J. Math. Biol., 72, 1441-1465 (2016) · Zbl 1338.35041 |
| [6] | Bownds, J. M.; Cushing, J. M., On the behaviour of solutions of predator-prey equations with hereditary terms, Math. Biosci., 26, 41-54 (1975) · Zbl 0333.92015 |
| [7] | Brauer, F.; Castillo-Chávez, C., Mathematical Models in Population Biology and Epidemiology (2012), Springer-Verlag: Springer-Verlag New York · Zbl 1302.92001 |
| [8] | Bürger, R.; Kumar, S.; Ruiz-Baier, R., Discontinuous finite volume element discretization for coupled flow-transport problems arising in models of sedimentation, J. Comput. Phys., 299, 446-471 (2015) · Zbl 1351.76098 |
| [9] | Bürger, R.; Ruiz-Baier, R.; Torres, H., A stabilized finite volume element formulation for sedimentation-consolidation processes, SIAM J. Sci. Comput., 34, B265-B289 (2012) · Zbl 1246.65177 |
| [10] | Burman, E.; Ern, A., Implicit-explicit Runge-Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations, ESAIM: Math. Model. Numer. Anal., 46, 681-707 (2012) · Zbl 1281.65123 |
| [11] | Cai, Z., On the finite volume element method, Numer. Math., 58, 713-735 (1991) · Zbl 0731.65093 |
| [12] | Chou, S. H., Analysis and convergence of a covolume method for the generalized Stokes problem, Math. Comp., 66, 85-104 (1997) · Zbl 0854.65091 |
| [13] | Chow, S.-N.; Hale, J. K., Methods of Bifurcation Theory (1982), Springer-Verlag: Springer-Verlag New York · Zbl 0487.47039 |
| [14] | Ciarlet, P. G., The Finite Element Method for Elliptic Problems (1978), North-Holland: North-Holland Amsterdam · Zbl 0383.65058 |
| [15] | Cross, M.; Greenside, H., Pattern Formation and Dynamics in Nonequilibrium Systems (2009), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1177.82002 |
| [16] | Cunningham, W.; Wangersky, P., Time lag in prey-predator population models, Ecology, 38, 136-139 (1957) |
| [17] | Dupraz, M.; Filippi, S.; Gizzi, A.; Quarteroni, A.; Ruiz-Baier, R., Finite element and finite volume-element simulation of pseudo-ECGs and cardiac alternans, Math. Methods Appl. Sci., 38, 6, 1046-1058 (2015) · Zbl 1309.65111 |
| [18] | Ewing, R. E.; Lazarov, R. D.; Lin, Y., Finite volume element approximations of nonlocal reactive flows in porous media, Numer. Methods Partial Differential Equations, 16, 285-311 (2000) · Zbl 0961.76050 |
| [19] | Freedman, H. I.; Hari Rao, V. S., The trade-off between mutual interference and time lags in predator-prey system, Bull. Math. Biol., 45, 991-1004 (1983) · Zbl 0535.92024 |
| [20] | Garvie, M. R., Finite-difference schemes for reaction-diffusion equations modeling predator-prey interactions in MATLAB, Bull. Math. Biol., 69, 931-956 (2007) · Zbl 1298.92081 |
| [21] | Garvie, M. R.; Trenchea, C., A three level finite element approximation of a pattern formation model in developmental biology, Numer. Math., 127, 397-422 (2014) · Zbl 1377.35154 |
| [22] | Gopalsamy, K., Pursuit evasion wave trains in prey-predator systems with diffusionally coupled delays, Bull. Math. Biol., 42, 871-887 (1980) · Zbl 0446.92018 |
| [23] | Gopalsamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics (1992), Kluwer: Kluwer Dordrecht · Zbl 0752.34039 |
| [24] | Gourley, S. A.; Liu, R.; Wu, J., Spatiotemporal patterns of disease spread: interaction of physiological structure, spatial movements, disease pogression and human intervention, (Magal, P.; Ruan, S., Structured Population Models in Biology and Epidemiology (2008), Springer-Verlag: Springer-Verlag Berlin), 165-208 |
| [25] | Gourley, S. A.; So, J.; Wu, J., Nonlocality of reaction-diffusion equations induced by delay: biological modeling and nonlinear dynamics, J. Math. Sci. (NY), 124, 5119-5153 (2004) · Zbl 1128.35360 |
| [26] | Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II (2002), Springer-Verlag: Springer-Verlag New York · Zbl 0994.65135 |
| [27] | Hale, J. K.; Koçak, H., Dynamics and Bifurcations (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0745.58002 |
| [28] | Hassard, B.; Kazarino, D.; Wan, Y., Theory and Applications of Hopf Bifurcation (1981), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0474.34002 |
| [29] | Huang, C., Delay-dependent stability of high order Runge-Kutta methods, Numer. Math., 111, 377-387 (2009) · Zbl 1167.65045 |
| [30] | Huang, C.; Vandewalle, S., Unconditionally stable difference methods for delay partial differential equations, Numer. Math., 122, 579-601 (2012) · Zbl 1272.65066 |
| [31] | Jana, S.; Chakraborty, M.; Chakraborty, K.; Kar, T. K., Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge, Math. Comput. Simul., 85, 57-77 (2012) · Zbl 1258.34161 |
| [32] | Koto, T., Stability of IMEX Runge-Kutta methods for delay differential equations, J. Comput. Appl. Math., 211, 201-212 (2008) · Zbl 1141.65065 |
| [33] | Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press New York · Zbl 0777.34002 |
| [34] | Li, J.; Chen, Z.; He, Y., A stabilized multi-level method for non-singular finite volume solutions of the stationary 3D Navier-Stokes equations, Numer. Math., 122, 279-304 (2012) · Zbl 1366.76062 |
| [35] | Lin, Z.; Ruiz-Baier, R.; Tian, C., Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion, J. Comput. Phys., 256, 806-823 (2014) · Zbl 1349.80036 |
| [36] | Malchow, H.; Petrovskii, S. V.; Venturino, E., Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, Simulations (2008), Chapman & Hall / CRC Press · Zbl 1298.92004 |
| [37] | May, R. M., Stability and Complexity in Model Ecosystems (1973), Princeton University Press: Princeton University Press Princeton |
| [38] | McKibben, M. A., Discovering Evolution Equations with Applications. Volume 1-Deterministic Equations (2011), CRC Press: CRC Press Boca Raton · Zbl 1362.35001 |
| [39] | Medvinsky, A. B.; Petrovskii, S. V.; Tikhonova, I. A.; Malchow, H.; Li, B.-L., Spatiotemporal complexity of plankton and fish dynamics, SIAM Rev., 44, 311-370 (2002) · Zbl 1001.92050 |
| [40] | Murray, J. D., Spatial structures in predator-prey communities-A nonlinear time delay diffusional model, Math. Biosci., 30, 73-85 (1976) · Zbl 0335.92002 |
| [41] | Murray, J. D., Mathematical Biology I: An Introduction (2002), Springer: Springer New York · Zbl 1006.92001 |
| [42] | Murray, J. D., Mathematical Biology II: Spatial Models and Biomedical Applications (2003), Springer: Springer New York · Zbl 1006.92002 |
| [43] | Nababan, S.; Teo, K. L., Existence and uniqueness of weak solutions of the Cauchy problem for parabolic delay-differential equations, Bull. Aust. Math. Soc., 21, 65-80 (1980) · Zbl 0418.35074 |
| [44] | Pao, C. V., Nonlinear Parabolic and Elliptic Equations (1992), Plenum Press: Plenum Press New York · Zbl 0777.35001 |
| [45] | Phongthanapanich, S.; Dechaumphai, P., Finite volume element method for analysis of unsteady reaction-diffusion problems, Acta Mech. Sin., 25, 481-489 (2009) · Zbl 1178.76252 |
| [46] | Quarteroni, A.; Ruiz-Baier, R., Analysis of a finite volume element method for the Stokes problem, Numer. Math., 118, 737-764 (2011) · Zbl 1230.65130 |
| [47] | Rodrigues, L. A.D.; Mistro, D. C.; Petrovskii, S., Pattern formation, long-term transients, and the Turing-Hopf bifurcation in a space- and time-discrete predator-prey system, Bull. Math. Biol., 73, 1812-1840 (2011) · Zbl 1220.92053 |
| [48] | Ruan, S., On nonlinear dynamics of predator-prey models with discrete delay, Math. Model. Nat. Phenom., 4, 140-188 (2009) · Zbl 1172.34046 |
| [49] | Ruan, S. G.; Wei, J. J., On the zero of some transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10, 863-874 (2003) · Zbl 1068.34072 |
| [50] | Sen, S.; Ghosh, P.; Riaz, S. S.; Ray, D. S., Time-delay-induced instabilities in reaction-diffusion systems, Phys. Rev. E, 80 (2009), paper 046212 |
| [51] | Shakeri, F.; Dehghan, M., The finite volume spectral element method to solve Turing models in the biological pattern formation, J. Comput. Math. Appl., 62, 12, 4322-4336 (2011) · Zbl 1236.65118 |
| [52] | Smith, H., An Introduction to Delay Differential Equations with Applications in the Life Sciences (2011), Springer-Verlag: Springer-Verlag New York · Zbl 1227.34001 |
| [53] | Sun, G.-Q.; Jin, Z.; Haque, M.; Li, B.-L., Spatial patterns of a predator-prey model with cross diffusion, Nonlinear Dynam., 69, 1631-1638 (2012) · Zbl 1263.34062 |
| [54] | Tian, C., Delay-driven spatial patterns in a plankton allelopathic system, Chaos, 22 (2012), paper 013129 · Zbl 1331.92150 |
| [55] | Volterra, V., Variazioni e fluttuazioni del numero d’individui in specie animali conviventi, Mem. Acad. Lincei, 2, 31-116 (1926) |
| [56] | Volterra, V., Leçons sur la Théorie Mathématique de la Lutte pour la Vie (1931), Gauthier-Villars: Gauthier-Villars Paris · Zbl 0002.04202 |
| [57] | Wang, W., Epidemic models with time delays, (Ma, Z.; Zhou, Y.; Wu, J., Modeling and Dynamics of Infectious Diseases (2009), Higher Education Press: Higher Education Press Beijing), 289-314 · Zbl 1180.92081 |
| [58] | Wang, W.; Chen, L. S., A predator-prey system with stage-structure for predator, J. Comput. Appl. Math., 33, 83-101 (1997) |
| [59] | Wang, Y.-M.; Pao, C. V., Time-delayed finite difference reaction-diffusion systems with nonquasimonotone functions, Numer. Math., 103, 485-513 (2006) · Zbl 1118.65096 |
| [60] | Xiao, A.; Zhang, G.; Zhou, J., Implicit-explicit time discretization coupled with finite element methods for delayed predator-prey competition reaction-diffusion system, Comput. Math. Appl., 71, 10, 2106-2123 (2016) · Zbl 1443.92171 |
| [61] | Zhang, J.-F.; Li, W.-T.; Yan, X.-P., Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator-prey models, Appl. Math. Comput., 218, 1883-1893 (2011) · Zbl 1228.92082 |
| [62] | Zhao, T.; Kuang, Y.; Smith, H. L., Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal., 28, 1373-1390 (1997) · Zbl 0872.34047 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
