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Kulakov M. P., Frisman E. Y. Approaches to study of multistability in spatio-temporal dynamics of two-age population. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2020, vol. 28, iss. 6, pp. 653-678. DOI: 10.18500/0869-6632-2020-28-6-653-678

# Approaches to study of multistability in spatio-temporal dynamics of two-age population

Purpose of the work is to study spatio-temporal dynamics of limited two-age structured populations that populate a 2D habitat and capable of long-range displacement of individuals. We proposed the model that is the network of nonlocally coupled nonlinear maps with nonlinear coupling function. Conditions for the emergence of different types of heterogeneous spatial distribution, combining coherent and incoherent regimes in different sites and solitary states are studied. Methods. In order to study the multistability of the dynamics in space and time, we used the synchronization factor and the order parameter. In addition, the method for estimating a number of solitary states is proposed. During numerical experiments, we generated many random initial conditions and computed these indicators for asymptotic space-time regime, and estimated the probability of a specific scenario. Results. Three typical regimes of spatio-temporal dynamics are described. The first one is a homogeneous distribution with full or partial synchronization. The probability of this scenario decreases as the strength and/or radius of coupling decreases. The second is a heterogeneous distribution with spots, stripes or labyrinths patterns, corresponding to cluster synchronization. The last one is highly fragmented spots, but in general with coherent dynamics. It was shown that these regimes coexist under certain conditions. Moreover, in most cases, the spatio-temporal dynamics contains randomly located single elements with outbreak of population size (solitary states) regardless of the observed regime of most others. Conclusion. The following paradoxical situation was revealed. As the elements become less coupled and the dynamics more incoherent, the number of solitary states increases. As a result, the elements with outbreak are more often synchronized with each other and form clusters of solitary states mixed with clusters of synchronous populations, or with highly fragmented clusters, or such clusters appear against the background of absolutely non-synchronous dynamics.

- Korneev I.A., Slepnev A.V., Semenov V.V., Vadivasova T.E. Wave processes in a ring of memristively coupled self-excited oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2020, vol. 28, iss. 3, pp. 324–340 (in Russian).
- Xu Y., Jia Y., Ma J., Alsaedi A., Ahmad B. Synchronization between neurons coupled by memristor. Chaos, Solitons & Fractals, 2017, vol. 104, pp. 435–442.
- Gonze D. Bernard S., Waltermann C., Kramer A., Herzel H. Spontaneous synchronization of coupled circadian oscillators. Biophysical Journal, 2005, vol. 89, no. 1, pp. 120–129.
- Shen Y. Hou Z., Xin H. Transition to burst synchronization in coupled neuron networks. Physical Review E, 2008, vol. 77, no. 031920, pp. 1–5.
- Ma J., Xu Y., Wang C. Jin W. Pattern selection and self-organization induced by random boundary initial values in a neuronal network. Physica A: Statistical Mechanics and its Applications, 2016, vol. 461, pp. 586–594.
- Peng M. et al. Multistability and complex dynamics in a simple discrete economic model. Chaos, Solitons & Fractals, 2009, vol. 41, no. 2, pp. 671–687.
- Volos C. K., Kyprianidis I. M., Stouboulos I. N. Synchronization phenomena in coupled nonlinear systems applied in economic cycles. WSEAS Trans. Syst., 2012, vol. 11, no. 12, pp. 681–690.
- Ikeda Y., Aoyama H., Yoshikawa H. Synchronization and the coupled oscillator model in international business cycles. RIETI Discussion Papers, 2013, no. 13-E-089.
- Earn D.J.D., Levin S.A., Rohani P. Coherence and Conservation. Science, 2000, vol. 290, no. 5495, pp. 1360–1364.
- Yakubu A.-A., Castillo-Chavez C. Interplay between local dynamics and disperal in discrete-time metapopulation model. Journal of Theoretical Biology, 2002, vol. 218, no. 3, pp. 273–288.
- Castro M.L., Silva J.A.L, Justo D.A.R. Stability in an age-structured metapopulation model. Journal of Mathematical Biology, 2006, vol. 52, no. 2, pp. 183–208.
- Wysham D.B., Hastings A. Sudden shift ecological systems: Intermittency and transients in the coupled Riker population model. Bulletin of Mathematical Biology, 2008, vol. 70, pp. 1013–1031.
- Silva J.A.L., Barrionuevo J.A., Giordani F.T. Synchronism in population networks with non linear coupling. Nonlinear Analysis: Real World Applications, 2009, vol. 11, no. 2, pp. 1005–1016.
- Kulakov M.P., Axenovich T.I., Frisman E.Ya. Approach to the description a spatial dynamics of migration-related populations. Regional problems, 2013, vol. 16, no 1, pp. 5–15 (in Russian).
- Kulakov M.P., Neverova G.P., Frisman E.Y. Multistability in dynamic models of migration coupled populations with an age structure. Russian Journal of Nonlinear Dynamic, 2014, vol. 10, no. 4, pp. 407–425 (in Russian).
- Kulakov M.P., Frisman E.Ya. Using clustering by coupled map lattices for metapopulation dynamics simulation. Mathematical Biology and Bioinformatics, 2015, vol. 10, no. 1, pp. 220–233 (in Russian).
- Kulakov M.P., Frisman E.Y. Clustering and chimeras in the model of the spatial-temporal dynamics of agestructured populations. Russian Journal of Nonlinear Dynamic, 2018, vol. 14, no. 1, pp. 13–31 (in Russian).
- Kulakov M.P., Frisman E.Y. Modeling the spatio-temporal dynamics of a population with age structure and long-range interactions: synchronization and clustering. Mathematical Biology and Bioinformatics, 2019, vol. 14, no 1. pp. 1–18 (in Russian).
- Ghorai S., Chakraborty P., Poria S. Bairagi N. Dispersal-induced pattern-forming instabilities in host–parasitoid metapopulations. Nonlinear Dynamics, 2020, vol. 100, pp. 749–762.
- Levin S.A. Dispersion and population interactions. The American Naturalist, 1974, vol. 108, no. 960, pp. 207–228.
- Logofet D.O. Is migration able to stabilize the ecosystem? (Mathematical aspect). Biology Bulletin Reviews, 1978, vol. 39, pp. 123–129 (in Russian).
- Fujisaka H., Yamada T. Stability theory of synchronized motion in coupled-oscillator systems. Progress of Theoretical Physics, 1983, vol. 69, no. 1, pp. 32–47.
- Yamada T., Fujisaka H. Stability theory of synchronized motion in coupled-oscillator systems. II: The mapping approach. Progress of Theoretical Physics, 1983, vol. 70, no. 5, pp. 1240–1248.
- Kaneko K. Transition from torus to chaos accompanied by frequency lockings with symmetry breaking: In connection with the coupled-logistic map. Progress of Theoretical Physics, 1983, vol. 69, no. 5, pp. 1427–1442.
- Kuznetsov S.P. Model description of a chain of coupled dynamic systems near order-disorder phase transitions. Soviet Physics Journal. 1984, vol. 27, pp. 522–530 (in Russian).
- Gyllenberg M., Soderbacka G., Ericson S. Does migration stabilize local population dynamics? ¨ Analysis of a discrete metapopulation model. Math. Biosciences, 1993, vol. 118, pp. 25–49.
- Udwadia F.E., Raju N. Dynamics of coupled nonlinear maps and its application to ecological modeling. Applied Mathematic and Computation, 1997, vol. 82, pp. 137–179.
- Oppo G.-L., Kapral R. Discrete models for the formation and evolution of spatial structure in dissipative systems. Phys. Rev. A, 1984, vol. 33, no. 6, pp. 4219–4231.
- Crutchfield J.P., Kaneko K. Phenomenology of spatio-temporal chaos. In book «Directions in Chaos – Volume 1». World Scientific Publishing Co. Pte. Ltd. 1987, pp. 272–353.
- Kaneko K. Clustering, coding, switching, hierarchical, ordering, and control in network of chaotic elements. Physica D, 1990, vol. 41, pp. 137–172.
- Kolmogorov A.N., Petrovsky I.G., Piskunov N.S. The study of the diffusion equation, coupled with an increase in the amount of substance, and its application to one biological problem. MSU Bulletin. Series A. Mathematics and Mechanics, 1937, vol. 6, no. 1, pp. 1–26 (in Russian).
- Fischer B.A. The wave of advance of advantageous genes. Ann. Eugenica, 1937, vol. 7, pp. 355–369.
- Turing A.M. The chemical basis of the morphogenesis. Phil. Trans. R. Soc. London B, 1952, vol. 237, pp. 37–71.
- Svirezhev Yu.M. Nonlinear Waves, Dissipative Structures and Catastrophes in Ecology. Nauka, Moscow, 1987 (in Russian).
- Belintsev B.N. Physical Foundations of Biological Morphogenesis. Nauka, Moscow, 1991 (in Russian).
- Koch A.J., Meinhardt H. Biological pattern formation: From basic mechanisms to complex structures. Rev. Mod. Phys, 1994, vol. 66, no. 1481.
- Li M., Han B., Xu L., Zhang G. Spiral patterns near Turing instability in a discrete reaction diffusion system. Chaos, Solitons & Fractals, 2013, vol. 49, pp. 1–6.
- Tyutyunov Yu.V., Titova L.I., Senina I.N. Prey-taxis destabilizes homogeneous stationary state in spatial Gause-Kolmogorov-type model for predator-prey system. Ecological Complexity, 2017, vol. 31, pp. 170–180.
- Vasconcelos D.B., Viana R.L., Lopes S.R., Batista A.M., Pinto S.E. de S. Spatial correlations and synchronization in coupled map lattices with long-range interactions. Physica A, 2004, vol. 343, pp. 201–218.
- Viana R.L., Batista A.M., Batista C.A.S., Iarosz K.C. Lyapunov spectrum of chaotic maps with a long-range coupling mediated by a diffusing substance. Nonlinear Dynamics, 2017, vol. 87, no. 3, pp. 1589–1601.
- Batista C.A.S., Viana R.L. Chaotic maps with nonlocal coupling: Lyapunov exponents, synchronization of chaos, and characterization of chimeras. Chaos, Solitons & Fractals, 2020, vol. 131, no. 109501.
- Frisman E.Y., Neverova G.P., Revutskaya O.L. Complex dynamics of the population with a simple age structure. Ecological Modelling, 2011, vol. 222, no. 12, pp. 1943–1950.
- Neverova G.P., Kulakov M.P., Frisman E.Y. Changes in population dynamics regimes as a result of both multistability and climatic fluctuation. Nonlinear Dynamics, 2019, vol. 97, no. 1, pp. 107–122.
- Zhang L., Zhang C. Codimension one and two bifurcations of a discrete stage-structured population model with self-limitation. Journal of Difference Equations and Applications, 2018, vol. 24, no. 8, pp. 1210–1246.
- Tuzinkevich A.V., Frisman E.Ya. Dissipative structures and patchiness in spatial distribution of plants. Ecological Modelling, 1990, vol. 52, pp. 207–223.
- Shepelev I.A., Vadivasova T.E., Bukh A.V., Strelkova G.I., Anishchenko V.S. New type of chimera structures in a ring of bistable FitzHugh–Nagumo oscillators with nonlocal interaction. Physics Letters A, 2017, vol. 381, no. 16, pp. 1398–1404.
- Rybalova E., Anishchenko V.S., Strelkova G.I., Zakharova A. Solitary states and solitary state chimera in neural networks. Chaos, 2019, vol. 29, no. 071106.
- Shepelev I.A., Bukh A.V., Vadivasova T.E., Anishchenko V.S., Zakharova A. Double-well chimeras in 2D lattice of chaotic bistable elements. Commun. Nonlinear. Sci. Numer. Simulat., 2018, vol. 54, pp. 50–61.
- Strelkova G.I., Anishchenko V.S. Spatio-temporal structures in ensembles of coupled chaotic systems. Advances in Physical Sciences, 2020, vol. 63, no. 2. pp. 160–178.
- Shepelev I.A., Vadivasova T.E. Solitary states in a 2D lattice of bistable elements with global and close to global interaction. Russian Journal of Nonlinear Dynamic, 2017, vol. 13, no. 3, pp. 317–329 (in Russian).
- Gopal R., Chandrasekar V.K., Venkatesan A., Lakshmanan M. Observation and characterization of chimera states in coupled dynamical systems with nonlocal coupling. Phys. Rev. E, 2014, vol. 89, no. 052914.
- Kuramoto Y., Nishikawa I. Statistical macrodynamics of large dynamical systems. Case of a phase transition in oscillator communities. Journal of Statistical Physics, 1987, vol. 49, no. 3–4, pp. 569–605.
- Restrepo J.G., Ott E., Hunt B.R. Onset of synchronization in large networks of coupled oscillators. Physical Review E, 2005, vol. 71, no. 036151.
- Hanski I.A., Gaggiotti O.E. (ed.). Ecology, Genetics and Evolution of Metapopulations. Academic Press, 2004.
- Barbosa P., Schultz J.C. Insect Outbreaks. Academic Press, Inc, 1987.
- Isaev A.S., Palnikova E.N., Sukhovolsky V.G., Tarasova O.V. Population Dynamics of Forest Phyllophagous Insects: Models and Forecasts. Moscow, Publ. KMK, 2015 (in Russian).

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