# 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:

show all references

##### References:
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
 [1] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [2] Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270 [3] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [4] Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 [5] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351 [6] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462 [7] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 [8] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [9] Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297 [10] Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272 [11] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [12] Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450 [13] Ebraheem O. Alzahrani, Muhammad Altaf Khan. Androgen driven evolutionary population dynamics in prostate cancer growth. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020426 [14] Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456 [15] Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383 [16] Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252 [17] Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078 [18] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 [19] Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457 [20] Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $\mathcal{W}(a, b, r)$. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

2019 Impact Factor: 1.27