| Citation: |
Yingwei Song, Tie Zhang. SPATIAL PATTERN FORMATIONS IN DIFFUSIVE PREDATOR-PREY SYSTEMS WITH NON-HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 165-177. doi: 10.11948/20190097
|
SPATIAL PATTERN FORMATIONS IN DIFFUSIVE PREDATOR-PREY SYSTEMS WITH NON-HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS
-
Abstract
A reaction-diffusion predator-prey system with non-homogeneous Dirichlet boundary conditions describes the persistence of predator and prey species on the boundary. Compared with homogeneous Neumann boundary conditions, the former conditions may prompt or prevent the spatial patterns produced through diffusion-induced instability. The spatial pattern formation induced by non-homogeneous Dirichlet boundary conditions is characterized by the Turing type linear instability of homogeneous state and bifurcation theory. Furthermore, transient spatiotemporal behaviors are observed through numerical simulations.-
Keywords:
- Reaction /
- diffusion /
- predator-prey /
- stability /
- bifurcation
-
-
References
[1] R. Arditi and L. R. Ginzburg, Coupling in predator-prey dynamics: Ratio-dependence, Journal of Theoretical Biology, 1989, 139(3), 311-326. doi: 10.1016/S0022-5193(89)80211-5 [2] R. Arditi, L. R. Ginzburg and H. R. Akcakaya, Variation in plankton densities among lakes: a case for ratio-dependent models, The American Naturalist, 1991, 138(5), 1287-1296. doi: 10.1086/285286 [3] R. Arditi, N. Perrin and H. Saiah, Functional responses and heterogeneities: an experimental test with cladocerans, Oikos., 1991, 60(1), 69-75. doi: 10.2307/3544994 [4] R. Arditi and H. Saiah, Empirical evidence of the role of heterogeneity in ratio-dependent consumption, Ecology, 1992, 73(5), 1544-1551. doi: 10.2307/1940007 [5] H. Beirao da Veiga, On the global regularity for singular p-systems under non-homogeneous dirichlet boundary conditions, Journal of Mathematical Analysis and Applications, 2013, 398, 527-533. doi: 10.1016/j.jmaa.2012.08.058 [6] P. A. Braza, The bifurcation structure of the holling-tanner model for predator-prey interactions using two-timing, SIAM J. Appl. Math., 2003, 63(3), 889-904. doi: 10.1137/S0036139901393494 [7] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Academic Press, New York, 2003. [8] C. Chen and L. Hung, Nonexistence of traveling wave solutions, exact and semi-exact traveling wave solutions for diffusive lotka-volterra systems of three competing species, Comm. Pure Appl Anal., 2017, 15(4), 1451-1469. [9] S. Chen, J. Shi and J. Wei, Global stability and hopf bifurcation in a delayed diffusive leslie-gower predator-prey system, International Journal of Bifurcation and Chaos, 2012. 10.1142/S0218127412500617. [10] L. Ciannelli, M. Hunsicker, M. Hidalgo et al., Theory, consequences and evidence of eroding population spatial structure in harvested marine fishes, Mar Ecol Prog Ser., 2013, 480, 227-243. doi: 10.3354/meps10067 [11] J. B. Collings, Bifurcation and stability analysis of a temperature-dependent mite predator-prey interaction model incorporating a prey refuge, Bull. Math. Biol., 1995, 57(1), 63-76. doi: 10.1016/0092-8240(94)00024-7 [12] Y. Du and S. B. Hsu, A diffusive predator-prey model in heterogeneous environment, Journal of Differential Equations, 2004, 203(2), 331-364. doi: 10.1016/j.jde.2004.05.010 [13] A. P. Gutierrez, Physiological basis of ratio-dependent predator-prey theory: the metabolic pool model as a paradigm, Ecology, 1992, 73(5), 1552-1563. doi: 10.2307/1940008 [14] J. Ha and S. Nakagiri, Damped sine-gordon equations with non-homogeneous dirichlet boundary conditions, Journal of Mathematical Analysis and Applications, 2001, 263(2), 708-720. [15] M. Han, Bifurcation theory and methods of dynamical systems, Science Press, Beijing, 1995. [16] M. P. Hassell, The dynamics of arthropod predator-prey systems, Monogr. Popul. Biol., 1978, 65(13), 1-237. [17] L. Hauser and G. R. Carvalho, Paradigm shifts in marine fisheries genetics: ugly hypotheses slain by beautiful facts, Fish Fish., 2010, 9(4), 333-362. [18] C. S. Holling, The functional response of predators to prey density and its role in mimicry and population regulation, Mem. Ent. Soc. Can., 1965, 97(45), 1-60. [19] C. S. Holling, The functional response of invertebrate predators to prey density, Mem. Ent. Soc. Can., 1966, 98(48), 1-86. [20] S. B. Hsu and T. W. Huang, Global stability for a class of predator-prey systems, SIAM J. Appl. Math., 1995, 55(3), 763-783. doi: 10.1137/S0036139993253201 [21] W. Ko and K. Ryu, Non-constant positive steady-states of a diffusive predator-prey system in homogeneous environment, J. Math. Anal. Appl., 2007, 327, 539-549. doi: 10.1016/j.jmaa.2006.04.077 [22] A. Korobeinikov, A lyapunov function for leslie-gower predator-prey models, Appl. Math. Lett., 2001, 14(6), 697-699. doi: 10.1016/S0893-9659(01)80029-X [23] Y. Lou and W. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 1996, 131, 79-131. doi: 10.1006/jdeq.1996.0157 [24] R. M. May, Limit cycles in predator-prey communities, Science, 1972, 177, 900-902. doi: 10.1126/science.177.4052.900 [25] R. M. May, Stability and Complexity in Model Ecosystems, Academic Press, New York, 1973. [26] R. Peng and M. Wang, Positive steady sates of the holling-tanner prey-predator model with diffusion, Proceedings of the Royal Society of Edinburgh, 2005, 135(1), 149-164. doi: 10.1017/S0308210500003814 [27] H. Reiss, G. Hoarau and M. Dickey-Collas, Genetic population structure of marine fish: mismatch between biological and fisheries management units, Fish Fish., 2009, 10(4), 361-395. doi: 10.1111/j.1467-2979.2008.00324.x [28] D. Robichaud and G. Rose, Migratory behaviour and range in atlantic cod: inference from a century of tagging, Fish Fish., 2004, 5(3), 185-214. doi: 10.1111/j.1467-2679.2004.00141.x [29] E. Saez and E. Gonzalez-Olivares, Dynamics of a predator-prey model, Siam Journal on Applied Mathematics, 1999, 59(5), 1867-1878. doi: 10.1137/S0036139997318457 [30] T. Saha and C. Chakrabarti, Dynamical analysis of a delayed ratio-dependent holling-tanner predator-prey model, Journal of Mathematical Analysis and Applications, 2009, 358(2), 389-402. doi: 10.1016/j.jmaa.2009.03.072 [31] D. E. Schindler, H. Ray and C. Brandon, Population diversity and the portfolio effect in an exploited species, Nature, 2010, 465(7298), 609-612. doi: 10.1038/nature09060 [32] S. M. Sohel Rana, Bifurcations and chaos control in a discrete-time predator-prey system of leslie type, Journal of Applied Analysis and Computation, 2019, 9(1), 31-44. [33] J. Song, M. Hu and Y. Bai, Dynamic analysis of a non-autonomous ratio-dependent predator-prey model with additional food, Journal of Applied Analysis and Computation, 2018, 8(6), 1893-1909. [34] N. Stenseth, P. Jorde and K. Chan, Ecological and genetic impact of atlantic cod larval drift in the skagerrak, Proc Roy Soc Biol., 2006, 273(1590), 1085-1092. doi: 10.1098/rspb.2005.3290 [35] J. T. Tanner, The stability and the intrinsic growth rates of prey and predator populations, Ecology, 1975, 56(4), 855-867. doi: 10.2307/1936296 [36] J. Wang, J. Shi and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong allee effect in prey, J. Differential Equations, 2011, 251(4), 1276-1304. [37] J. Wang, J. Shi and J. Wei, Global bifurcation analysis and pattern formation in homogeneous diffusive predator-prey systems, J. Differential Equations, 2016, 260(4), 3495-3523. doi: 10.1016/j.jde.2015.10.036 [38] L. Wang, J. Watmough and F. Yu, Bifurcation analysis and transient spatio-temporal dynamics for a diffusive plant-herbivore system with dirichlet boundary conditions, Mathematical Biosciences and Engineering, 2015, 12(4), 699-715. doi: 10.3934/mbe.2015.12.699 [39] D. J. Wollkind, J. B. Collings and J. A. Logan, Metastability in a temperature-dependent model system for predator-prey mite outbreak interactions on fruit trees, Bull. Math. Biol., 1988, 50(4), 379-409. doi: 10.1016/S0092-8240(88)90005-5 [40] S. Wu, J. Wang and J. Shi, Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models. Methods. Appl. Sci., 2018, 28(11), 2275-2312. doi: 10.1142/S0218202518400158 [41] Y. Yamada, Stability of steady states for prey-predator diffusion equations with homogeneous dirichlet conditions, SIAM J. Math. Anal., 1990, 21(2), 327-345. doi: 10.1137/0521018 [42] F. Yi, J. Wei and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system, J. Diff. Eqs., 2009, 246(4), 1944-1977. -
-
Figures(6)
Export File
Citation
Yingwei Song, Tie Zhang. SPATIAL PATTERN FORMATIONS IN DIFFUSIVE PREDATOR-PREY SYSTEMS WITH NON-HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS[J]. Journal of Applied Analysis & Computation, 2020, 10(1): 165-177. doi: 10.11948/20190097
Format
Content
DownLoad:
-
Figure 1. The
$ u $ -component and$ v $ -component of a numerical solution of system (1.1) with non-homogeneous Dirichlet boundary conditions. Parameter values:$ p = 0.3, a = 0.1, b = 0.3\times10^{-14}, d_1 = 1000, d_2 = 0.1 $ and the initial condition:$ (u_0(x), v_0(x)) = (0.0185+0.005\sin(4x), 0.0185+0.003\sin(6x)) $ . The positive equilibrium$ (u^{*}, v^{*}) $ is stable and the solution converges to$ (u^{*}, v^{*}) = (0.0185, 0.0185) $ -
Figure 2. The
$ u $ -component and$ v $ -component of a numerical solution of system (1.1) with homogeneous Neumann boundary conditions. Parameter values:$ p = 0.3, a = 0.1, b = 0.3\times10^{-14}, d_{1} = 1000, d_{2} = 0.1 $ and initial condition:$ (u_0(x), v_0(x)) = (0.0185+0.005\sin(4x), 0.0185+0.003\sin(6x)) $ . When only the boundary conditions are changed and the same parameters and initial value are kept the same as in Figure 1, the solution does not converge to$ (u^{*}, v^{*}) = (0.0185, 0.0185) $ -
Figure 3. The
$ u $ -component and$ v $ -component of a numerical solution of system (1.1) with parameter values:$ \sigma = 0.065, p = 0.8, a = 0.1, b = 0.3, d_{1} = 1, d_{2} = 0.7 $ and the initial condition:$ (u_0(x), v_0(x)) = (0.2598+0.005\sin(4x), 0.2598+0.003\sin(6x)) $ . Since$ \sigma = 0.065 > \sigma(\overline{\mu_{i}}) = 0.064 $ , the positive equilibrium$ (u^{*}, v^{*}) = (0.2598, 0.2598) $ is stable -
Figure 4. The
$ u $ -component and$ v $ -component of a numerical solution of system (1.1) with parameter values:$ \sigma = 0.063, p = 0.8, a = 0.1, b = 0.3, d_{1} = 1, d_{2} = 0.7 $ and the initial condition:$ (u_0(x), v_0(x)) = (0.2598+0.005\sin(4x), 0.2598+0.003\sin(6x)) $ . Since$ \sigma = 0.063 < \sigma(\overline{\mu_{i}}) = 0.064 $ , steady state bifurcation occurs resulting non-constant steady states -
Figure 5. The
$ u $ -component and$ v $ -component of a numerical solution of system (1.1) with parameter values:$ \sigma = 0.51, p = 0.8, a = 0.1, b = 0.3, d_{1} = 0.002, d_{2} = 0.001 $ and the initial condition:$ (u_0(x), v_0(x)) = (0.2598+0.005\sin(4x), 0.2598+0.003\sin(6x)) $ . Since$ \sigma = 0.51 > \sigma_{1}^{H} = 0.4427 $ , the positive equilibrium$ (u^{*}, v^{*}) $ is stable and the solution converges to$ (u^{*}, v^{*}) = (0.2598, 0.2598) $ -
Figure 6. The
$ u $ -component and$ v $ -component of a numerical solution of system (1.1) with parameter values:$ \sigma = 0.4426, p = 0.8, a = 0.1, b = 0.3, d_{1} = 0.002, d_{2} = 0.001 $ and the initial condition:$ (u_0(x), v_0(x)) = (0.2598+0.1\sin(x), 0.2598+0.1\sin(x)) $ . Since$ \sigma = 0.4426 $ is near the Hopf bifurcation value$ \sigma_{1}^{H} = 0.4427 $ , the solution converges to a spatially non-homogeneous periodic orbit