# American Institute of Mathematical Sciences

February 2019, 24(2): 547-561. doi: 10.3934/dcdsb.2018196

## Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat

 1 Don State Technical University, Rostov-on-Don, 344002, Russia 2 University of Rostock, D-18051 Rostock, Germany 3 Southern Federal University, Rostov-on-Don, 344090, Russia

* Corresponding author

Received  November 2016 Revised  April 2018 Published  June 2018

Fund Project: AVB and VGT are supported by Russian Foundation for Basic Research (grant 18-01-00453)

We explore an approach based on the theory of cosymmetry to model interaction of predators and prey in a two-dimensional habitat. The model under consideration is formulated as a system of nonlinear parabolic equations with spatial heterogeneity of resources and species. Firstly, we analytically determine system parameters, for which the problem has a nontrivial cosymmetry. To this end, we formulate cosymmetry relations. Next, we employ numerical computations to reveal that under said cosymmetry relations, a one-parameter family of steady states is formed, which may be characterized by different proportions of predators and prey. The numerical analysis is based on the finite difference method (FDM) and staggered grids. It allows to follow the transformation of spatial patterns with time. Eventually, the destruction of the continuous family of equilibria due to mistuned parameters is analyzed. To this end, we derive the so-called cosymmetric selective equation. Investigation of the selective equation gives an insight into scenarios of local competition and coexistence of species, together with their connection to the cosymmetry relations. When the cosymmetry relation is only slightly violated, an effect we call 'memory on the lost family' may be observed. Indeed, in this case, a slow evolution takes place in the vicinity of the lost states of equilibrium.

Citation: Alexander V. Budyansky, Kurt Frischmuth, Vyacheslav G. Tsybulin. Cosymmetry approach and mathematical modeling of species coexistence in a heterogeneous habitat. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 547-561. doi: 10.3934/dcdsb.2018196
##### References:
 [1] M. Banegje and S. Petrovski, Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system, J. Theor. Biol., 4 (2011), 37-53. [2] P. Begon, J. L. Harper and P. R. Townsend, Ecology: Individuals, Populations and Communities, Blackwell Scientific Publications Oxford, 1986. [3] A. V. Budyansky and V. G. Tsybulin, Impact of directed migration on formation of spatial structures of populations, Biophysics, 60 (2015), 622-631. [4] C. Cosner, Beyond diffusion: Conditional dispersal in ecological models, In: Infinite dimensional dynamical systems. Fields Institute Commun., 64 (2013), 305-317. doi: 10.1007/978-1-4614-4523-4_12. [5] C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 1701-1745. doi: 10.3934/dcds.2014.34.1701. [6] C. Cosner and R. Cantrell, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley and Sons Ltd, Chichester. 2003. doi: 10.1002/0470871296. [7] A. V. Epifanov and V. G. Tsybulin, Modeling of oscillatory scenarios of the coexistence of competing populations, Biophysics, 61 (2016), 696-704. doi: 10.1134/S0006350916040072. [8] A. R., Fisher The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. [9] K. Frischmuth and V. G. Tsybulin, Families of equilibria and dynamics in a population kinetics model with cosymmetry, Phys. Lett. A, 338 (2005), 51-59. doi: 10.1016/j.physleta.2005.02.015. [10] K. Frischmuth, E. S. Kovaleva and V. G. Tsybulin, Family of equilibria in a population kinetics model and its collapse, Nonlinear Analysis: Real World Applications, 12 (2011), 146-155. doi: 10.1016/j.nonrwa.2010.06.004. [11] G. F. Gause, The struggle for existence, Soil Science, 41 (1936), p159. doi: 10.1097/00010694-193602000-00018. [12] R. Gejji, Y. Lou, D. Munther and J. Peyton, Evolutionary convergence to ideal free dispersal strategies and coexistence, Bull. Math. Biol., 74 (2012), 257-299. doi: 10.1007/s11538-011-9662-4. [13] V. N. Govorukhin, A. B. Morgulis and Yu. V. Tyutyunov, Slow taxis in a predator-prey model, Dokl. Math., 61 (2000), 420-422; (Translated from Dokl. Akad. Nauk, 372 (2000), 730-732). [14] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [15] E. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. [16] A. Kolmogorov, I. Petrovskii and N. Piscounov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ., Math. Mech., 1 (1937), 1-25. [17] L. Korobenko and E. Braverman, On logistic models with a carrying capacity dependent diffusion: Stability of equilibria and coexistence with a regularly diffusing population, Nonlinear Analysis: Real World Applications, 13 (2012), 2648-2658. doi: 10.1016/j.nonrwa.2011.12.027. [18] K.-Y. Lam and Y. Lou, Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal, Bull. Math. Biol., 76 (2014), 261-291. doi: 10.1007/s11538-013-9901-y. [19] K.-Y. Lam, Y. Lou and F. Frithjof Lutscher, The emergence of range limits in advective environments, SIAM J. Appl. Math., 76 (2016), 641-662. doi: 10.1137/15M1027887. [20] Y. Lou, D. Xiao and P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discrete and Continuous Dynamical Systems - Series A, 36 (2016), 953-969. doi: 10.3934/dcds.2016.36.953. [21] M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments, Am. Nat., 140 (1992), 1010-1027. doi: 10.1086/285453. [22] J. D. Murray, Mathematical Biology Ⅱ. Spatial models and Biomedical Applications, Springer-Verlag, 2003. [23] V. Volterra, Leçons Sur la Théorie Mathématique de la Lutte Pour La Vie, Éditions Jacques Gabay, Sceaux, 1990. [24] L. Xue, Pattern formation in a predator-prey model with spatial effects, Physica A., 391 (2012), 5987-5996. doi: 10.1016/j.physa.2012.06.029. [25] V. I. Yudovich, Cosymmetry, degeneracy of the solutions of operator equations and the onset of filtrational convection, Math. Notes, 49 (1991), 540-545. doi: 10.1007/BF01142654. [26] V. I. Yudovich, Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it, Chaos, 5 (1995), 402-411. doi: 10.1063/1.166110. [27] V. I. Yudovich, Bifurcations under perturbations violating cosymmetry, Physics-Doklady, 49 (2004), 522-526. doi: 10.1134/1.1810578. [28] X.-C. Zhang, G.-Q. Sun and Z. Jin, Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response, Dynam. Systems Appl., 20 (2011), 1-15.

show all references

##### References:
 [1] M. Banegje and S. Petrovski, Self-organised spatial patterns and chaos in a ratio-dependent predator-prey system, J. Theor. Biol., 4 (2011), 37-53. [2] P. Begon, J. L. Harper and P. R. Townsend, Ecology: Individuals, Populations and Communities, Blackwell Scientific Publications Oxford, 1986. [3] A. V. Budyansky and V. G. Tsybulin, Impact of directed migration on formation of spatial structures of populations, Biophysics, 60 (2015), 622-631. [4] C. Cosner, Beyond diffusion: Conditional dispersal in ecological models, In: Infinite dimensional dynamical systems. Fields Institute Commun., 64 (2013), 305-317. doi: 10.1007/978-1-4614-4523-4_12. [5] C. Cosner, Reaction-diffusion-advection models for the effects and evolution of dispersal, Discrete and Continuous Dynamical Systems - Series A, 34 (2014), 1701-1745. doi: 10.3934/dcds.2014.34.1701. [6] C. Cosner and R. Cantrell, Spatial Ecology Via Reaction-Diffusion Equations, John Wiley and Sons Ltd, Chichester. 2003. doi: 10.1002/0470871296. [7] A. V. Epifanov and V. G. Tsybulin, Modeling of oscillatory scenarios of the coexistence of competing populations, Biophysics, 61 (2016), 696-704. doi: 10.1134/S0006350916040072. [8] A. R., Fisher The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), 353-369. [9] K. Frischmuth and V. G. Tsybulin, Families of equilibria and dynamics in a population kinetics model with cosymmetry, Phys. Lett. A, 338 (2005), 51-59. doi: 10.1016/j.physleta.2005.02.015. [10] K. Frischmuth, E. S. Kovaleva and V. G. Tsybulin, Family of equilibria in a population kinetics model and its collapse, Nonlinear Analysis: Real World Applications, 12 (2011), 146-155. doi: 10.1016/j.nonrwa.2010.06.004. [11] G. F. Gause, The struggle for existence, Soil Science, 41 (1936), p159. doi: 10.1097/00010694-193602000-00018. [12] R. Gejji, Y. Lou, D. Munther and J. Peyton, Evolutionary convergence to ideal free dispersal strategies and coexistence, Bull. Math. Biol., 74 (2012), 257-299. doi: 10.1007/s11538-011-9662-4. [13] V. N. Govorukhin, A. B. Morgulis and Yu. V. Tyutyunov, Slow taxis in a predator-prey model, Dokl. Math., 61 (2000), 420-422; (Translated from Dokl. Akad. Nauk, 372 (2000), 730-732). [14] T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217. doi: 10.1007/s00285-008-0201-3. [15] E. Keller and L. A. Segel, Model for chemotaxis, J. Theor. Biol, 30 (1971), 225-234. doi: 10.1016/0022-5193(71)90050-6. [16] A. Kolmogorov, I. Petrovskii and N. Piscounov, A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem, Bull. Moscow Univ., Math. Mech., 1 (1937), 1-25. [17] L. Korobenko and E. Braverman, On logistic models with a carrying capacity dependent diffusion: Stability of equilibria and coexistence with a regularly diffusing population, Nonlinear Analysis: Real World Applications, 13 (2012), 2648-2658. doi: 10.1016/j.nonrwa.2011.12.027. [18] K.-Y. Lam and Y. Lou, Evolutionarily stable and convergent stable strategies in reaction-diffusion models for conditional dispersal, Bull. Math. Biol., 76 (2014), 261-291. doi: 10.1007/s11538-013-9901-y. [19] K.-Y. Lam, Y. Lou and F. Frithjof Lutscher, The emergence of range limits in advective environments, SIAM J. Appl. Math., 76 (2016), 641-662. doi: 10.1137/15M1027887. [20] Y. Lou, D. Xiao and P. Zhou, Qualitative analysis for a Lotka-Volterra competition system in advective homogeneous environment, Discrete and Continuous Dynamical Systems - Series A, 36 (2016), 953-969. doi: 10.3934/dcds.2016.36.953. [21] M. A. McPeek and R. D. Holt, The evolution of dispersal in spatially and temporally varying environments, Am. Nat., 140 (1992), 1010-1027. doi: 10.1086/285453. [22] J. D. Murray, Mathematical Biology Ⅱ. Spatial models and Biomedical Applications, Springer-Verlag, 2003. [23] V. Volterra, Leçons Sur la Théorie Mathématique de la Lutte Pour La Vie, Éditions Jacques Gabay, Sceaux, 1990. [24] L. Xue, Pattern formation in a predator-prey model with spatial effects, Physica A., 391 (2012), 5987-5996. doi: 10.1016/j.physa.2012.06.029. [25] V. I. Yudovich, Cosymmetry, degeneracy of the solutions of operator equations and the onset of filtrational convection, Math. Notes, 49 (1991), 540-545. doi: 10.1007/BF01142654. [26] V. I. Yudovich, Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it, Chaos, 5 (1995), 402-411. doi: 10.1063/1.166110. [27] V. I. Yudovich, Bifurcations under perturbations violating cosymmetry, Physics-Doklady, 49 (2004), 522-526. doi: 10.1134/1.1810578. [28] X.-C. Zhang, G.-Q. Sun and Z. Jin, Spatial dynamics in a predator-prey model with Beddington-DeAngelis functional response, Dynam. Systems Appl., 20 (2011), 1-15.
Two members of the family of stationary distributions of species; $\widehat{\alpha}_{1} = \widehat{\alpha}_{2} = \widehat{\beta}_{12} = \widehat{\beta}_{21} = 0$
Steady density distributions of populations $u$, $v$, $w$ for $\widehat{\alpha}_{1} = \widehat{\alpha}_{2} = 0.2$, $\widehat{\beta}_{12} = \widehat{\beta}_{21} = 0$ (left column) and $\widehat{\alpha}_{1} = \widehat{\alpha}_{2} = 0.2$, $\widehat{\beta}_{12} = \widehat{\beta}_{21} = -0.1$ (right column)
Families of stationary distributions of prey for different predator coefficients: $\mu_{31} = 1.4$, $\mu_{32} = 1.4$ (curve 1); $\mu_{31} = 0.8$, $\mu_{32} = 1.4$ (2); $\mu_{31} = 1.4$, $\mu_{32} = 0.8$ (3); a family for the system without predator (line $PQ$)
Selective functions for different values of migration parameters: $\beta_{12} = \beta_{21} = 0.08$ (curve 1), $\beta_{12} = \beta_{21} = -0.08$ (2), $\beta_{12} = \beta_{21} = -0.06$ (3), $\beta_{12} = 0.08$, $\beta_{21} = -0.08$ (4), $\theta$ - index of the family member
Evolution to isolated equilibria from the family (dotted line) in the case of cosymmetry destruction: $\beta_{12} = \beta_{21} = 0.06$ (1), $\beta_{12} = \beta_{21} = 0.08$ (2)
Map of migration parameters, corresponding to coexistence (Ⅲ) or survival of only one of the species, $u$ (Ⅰ) or $v$ (Ⅱ), without predator (top) and with predator (bottom). The thick line indicates the existence of a continuous family of stationary states
Migration parameters, mean values of the densities $\overline{U}$, $\overline{V}$ and $\overline{W}$, elements of the stability spectrum of stationary solutions
 No. $\widehat{\alpha}_{1}$ $\widehat{\beta}_{12}$ $\widehat{\beta}_{21}$ $\overline{U}$ $\overline{V}$ $\overline{W}$ spectra 1 0 0 0 0.11 0.48 0.24 -2:2·10-6 -0.06 -0.37 Fig. 1A 2 0 0 0 0.22 0.24 0.45 -9:8·10-7 -0.07 -0.39 Fig. 1B 3 0.2 0 0 0.06 0.26 0.03 -0.03 -0.14 -0.41 Fig. 2C 4 0.2 -0.1 -0.1 0.07 0.26 0.04 -0.06 -0.37 -0.42 Fig. 2D
 No. $\widehat{\alpha}_{1}$ $\widehat{\beta}_{12}$ $\widehat{\beta}_{21}$ $\overline{U}$ $\overline{V}$ $\overline{W}$ spectra 1 0 0 0 0.11 0.48 0.24 -2:2·10-6 -0.06 -0.37 Fig. 1A 2 0 0 0 0.22 0.24 0.45 -9:8·10-7 -0.07 -0.39 Fig. 1B 3 0.2 0 0 0.06 0.26 0.03 -0.03 -0.14 -0.41 Fig. 2C 4 0.2 -0.1 -0.1 0.07 0.26 0.04 -0.06 -0.37 -0.42 Fig. 2D
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