Complex dynamics modes in a simple model of prey-predator community: Bistability and multistability

G. P. Neverova ; O. L. Zhdanova

Matematičeskaâ biologiâ i bioinformatika, Tome 18 (2023) no. 2, pp. 308-322.

Résumé

The paper proposes and studies a two-component discrete-time model of the prey-predator community considering zooplankton and fish interactions and their development features. Discrete-time systems of equations allow us to take into account naturally the rhythm of many processes occurring in marine and freshwater communities, which are subject to cyclical fluctuations due to the daily and seasonal cycle. We describe the dynamics of fish and zooplankton populations constituting the community by Ricker’s model, which is well-studied and widely used in population modeling. To consider the species interaction, we use the Holling-II type response function taking into account predator saturation. We carried out the study of the proposed model. The system is shown to have from one to three non-trivial equilibria, which gives the existence of the complete community. In addition to the saddle-node bifurcation, which generates bistability of stationary dynamics, a nontrivial equilibrium loses its stability according to the Neimark–Sacker scenario with an increase in the reproductive potential of both predator and prey species, as a result of which the community exhibits long-period oscillations similar to those observed in experiments. With the higher bifurcation parameter, the reverse Neimark–Sacker bifurcation is shown to occur followed by the closed invariant curve collapses, and dynamics of the population stabilizes, later losing stability through a cascade of period-doubling bifurcations. Multistability complicates the birth and disappearance of the invariant curve in the phase space scenario by the emergence of another irregular dynamics in the system with the single unstable nontrivial fixed point. At fixed values of the model parameters and different initial conditions, the system considered is shown to demonstrate various quasi-periodic oscillations. Despite extreme simplicity, the proposed discrete-time model of community dynamics demonstrates a wide variety and variability of dynamic modes. It shows that the influence of environmental conditions can change the type and nature of the observed dynamics.

Bibliographie

[1] V. P. Shuntov, Biologiya dalnevostochnykh morei Rossii, TINRO tsentr, Vladivostok, 2016, 604 pp.

[2] K. M. Gorbatenko, “Raspredelenie, biomassa, mezhgodovaya dinamika sagitt Okhotskogo morya”, Izv. TINRO, 184 (2016), 168–177

[3] E. P. Dulepova, “Dinamika produktsionnykh pokazatelei zooplanktona v severo zapadnoi chasti Beringova morya v sovremennyi period”, Izv. TINRO, 187 (2016), 187–196

[4] E. P. Dulepova, “Sostoyanie planktonnykh soobschestv i kormovaya obespechennost mintaya v severo-zapadnoi chasti Beringova morya v sovremennyi period”, Tr. VNIRO, 174 (2018), 91–104

[5] E. P. Dulepova, Sravnitelnaya bioproduktivnost makroekosistem dalnevostochnykh morei, TINRO-tsentr, Vladivostok, 2002, 274 pp.

[6] V. A. Silkin, A. I. Abakumov, L. A. Pautova, S. V. Pakhomova, A. V. Lifanchuk, “Mechanisms of regulation of invasive processes in phytoplankton on the example of the north-eastern part of the Black Sea”, Aquatic Ecology, 50:2 (2016), 221–234 | DOI

[7] S. G. Leles, J. E.L. Valentin, G. M. Figueiredo, “Evaluation of the complexity and performance of marine planktonic trophic models”, Anais da Academia Brasileira de Ciencias, 88 (2016), 1971–1991 | DOI

[8] S. V. Berdnikov, V. V. Selyutin, F. A. Surkov, Yu. V. Tyutyunov, “Modelirovanie morskikh ekosistem: opyt, sovremennye podkhody, napravleniya razvitiya (obzor). Chast 2. Modeli populyatsii i trofodinamiki”, Morskoi gidrofizicheskii zhurnal, 38:2 (2022), 196–217 | DOI | MR

[9] M. Scheffer, S. Rinaldi, Y. A. Kuznetsov, E. H. van Nes, “Seasonal dynamics of Daphnia and algae explained as a periodically forced predator-prey system”, Oikos, 1997, 519–532 | DOI

[10] E. Steffen, H. Malchow, A. B. Medvinsky, “Effects of seasonal perturbations on a model plankton community”, Environmental Modeling Assessment, 2 (1997), 43–48 | DOI

[11] A. B. Medvinskii, S. V. Petrovskii, I. A. Tikhonova, D. A. Tikhonov, B. L. Li, E. Venturino, G. R. Ivanitskii, “Formirovanie prostranstvenno-vremennykh struktur, fraktaly i khaos v kontseptualnykh ekologicheskikh modelyakh na primere dinamiki vzaimodeistvuyuschikh populyatsii planktona i ryby”, Uspekhi fizicheskikh nauk, 172:1 (2002), 31–66 | DOI

[12] S. V. Petrovskii, H. Malchow, “A minimal model of pattern formation in a prey-predator system”, Mathematical and Computer Modelling, 29:8 (1999), 49–63 | DOI | MR | Zbl

[13] S. V. Petrovskii, H. Malchow, “Wave of chaos: new mechanism of pattern formation in spatio-temporal population dynamics”, Theoretical Population Biology, 59:2 (2001), 157–174 | DOI | Zbl

[14] M. Scheffer, S. Rinaldi, Y. A. Kuznetsov, “Effects of fish on plankton dynamics: a theoretical analysis”, Canadian Journal of Fisheries and Aquatic Sciences, 57:6 (2000), 1208–1219 | DOI

[15] A. B. Medvinsky, I. A. Tikhonova, R. R. Aliev, B. L. Li, Z. S. Lin, H. Malchow, “Patchy environment as a factor of complex plankton dynamics”, Physical Review E, 64:2 (2001), 021915 | DOI | MR

[16] G. P. Neverova, O. L. Zhdanova, A. I. Abakumov, “Diskretnaya model sezonnogo tsveteniya planktona”, Matematicheskaya biologiya i bioinformatika, 15:2 (2020), 235–250 | DOI

[17] A. D. Bazykin, Matematicheskaya biofizika vzaimodeistvuyuschikh populyatsii, Nauka, M., 1985, 182 pp.

[18] Yu. M. Svirezhev, “Nonlinearities in mathematical ecology: Phenomena and models”, Would we live in Volterra's world? Ecological Modelling, 216 (2008), 89–101 | DOI

[19] Yu. V. Tyutyunov, L. I. Titova, F. A. Surkov, E. N. Bakaeva, “Troficheskaya funktsiya kolovratok-fitofagov (Rotatoria). Eksperiment i modelirovanie”, Zhurnal obschei biologii, 71:1 (2010), 52–62

[20] Yu. V. Tyutyunov, L. I. Titova, “Ot Lotki-Volterra k Arditi-Ginzburgu: 90 let evolyutsii troficheskikh funktsii”, Zhurnal obschei biologii, 79:6 (2018), 428–448 | DOI

[21] A. B. Medvinskii, B. V. Adamovich, A. V. Rusakov, D. A. Tikhonov, N. I. Nurieva, V. M. Tereshko, “Dinamika populyatsii: matematicheskoe modelirovanie i realnost”, Biofizika, 64:6 (2019), 1169–1192 | DOI

[22] G. P. Neverova, O. L. Zhdanova, “Sravnitelnyi analiz dinamiki prostykh matematicheskikh modelei planktonnogo soobschestva c razlichnymi funktsiyami otklika”, Matematicheskaya biologiya i bioinformatika, 17:2 (2022), 465–480 | DOI | MR

[23] U. E. Riker, Metody otsenki i interpretatsii biologicheskikh pokazatelei populyatsii ryb, Pisch. prom-t, M., 1979

[24] E. Ya. Frisman, M. P. Kulakov, O. L. Revutskaya, O. L. Zhdanova, G. P. Neverova, “Osnovnye napravleniya i obzor sovremennogo sostoyaniya issledovanii dinamiki strukturirovannykh i vzaimodeistvuyuschikh populyatsii”, Kompyuternye issledovaniya i modelirovanie, 11:1 (2019), 119–151 | DOI | MR

[25] E. Ya. Frisman, O. L. Zhdanova, M. P. Kulakov, G. P. Neverova, O. L. Revutskaya, “Matematicheskoe modelirovanie populyatsionnoi dinamiki na osnove rekurrentnykh uravnenii: rezultaty i perspektivy. Chast I”, Izvestiya RAN. Seriya biologicheskaya, 2021, no. 1, 3–18 | DOI

[26] A. P. Kuznetsov, Yu. V. Sedova, “Bifurkatsii trekhmernykh i chetyrekhmernykh otobrazhenii: universalnye svoistva”, Izvestiya vysshikh uchebnykh zavedenii. Prikladnaya nelineinaya dinamika, 20:5 (2012), 26–43 | DOI | Zbl

[27] A. P. Kuznetsov, A. V. Savin, Yu. V. Sedova, L. V. Tyuryukina, Bifurkatsii otobrazhenii, OOO Izdatelskii tsentr «Nauka», Saratov, 2012, 196 pp.

[28] M. Kulakov, G. Neverova, E. Frisman, “The Ricker competition model of two species: dynamic modes and phase multistability”, Mathematics, 10:7 (2022), 1076 | DOI