Abstract
We study a model of the active continuous media driven by the interspecific taxis and also by an external signal. We employ the Patlak-Keller-Segel law for modelling the tactical motions. We address the short-wavelength signals with the use the homogenization. We show that such a signal exerts the effect on the large-scale dynamics by the drift that arises from the homogenization. We analyse in detail the signals which have the form of travelling waves. We find out that they are capable of producing the quasi-equilibria—that is, the short-wavelength patterns which stay in equilibrium on average. We examine the stability of the quasi-equilibria and compare the results to the case when the signal is off. The effect of the signal turns out to be not single-valued but depending on the speed at which the signal-producing wave propagates. Namely, there is an independent threshold value such that increasing the amplitude of the wave destabilizes the quasi-equilibria provided that the wave speed is above this value. Otherwise, the same action exerts the opposite effect. It is worth to note that the effect is exponential in the amplitude of the signal in both cases.
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Notes
- 1.
While considering the signals (6), we require 2π∕c-periodicity in τ instead of 2π −periodicity. In accordance with this, we re-define the averaging, 〈⋅〉.
- 2.
Given the cosymmetry, the limit cycle generically does not branch off from the critical equilibrium except for integrable cosymmetry that is equivalent to a conservation law. In case of such an exception, the cycle branches off ‘as usual’ provided that there is no additional degeneration.
References
Bellomo, N., Bellouquid, A., Tao, Y. et al.: Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25(09), 1663–1763 (2015)
Govorukhin, V., Morgulis, A., Tyutyunov, Y.: Slow taxis in a predator-prey model. Doklady Math. 61(3), 420–422 (2000)
Arditi, R., Tyutyunov, Y., Morgulis, A., et al.: Directed movement of predators and the emergence of density-dependence in predator–prey models. Theor. Popul Biol. 59(3), 207–221 (2001)
Pearce, I.G., Chaplain, M.A.J., Schofield, P.G., et al.: Chemotaxis-induced spatio-temporal heterogeneity in multi-species host-parasitoid systems. J. Math. Biol. 55(3), 365–388 (2007)
Tyutyunov, Y., Titova, L., Senina, I.: Prey-taxis destabilizes homogeneous stationary state in spatial Gause–Kolmogorov-type model for predator–prey system. Ecol. Complexity 31, 170–180 (2017)
Wang, Q., Yang, J., Zhang, L.: Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: effect of cellular growth. Discrete Contin. Dyn. Syst. B 22(9), 3547–3574 (2017)
Allaire, G.: A brief introduction to homogenization and miscellaneous applications. In: ESAIM Proceedings, vol. 37. EDP Sciences, France (2012)
Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518 (1992)
Black, T.: Boundedness in a Keller–Segel system with external signal production. J. Math. Anal. Appl. 446(1), 436–455 (2017)
Issa, T.B., Shen, W.: Persistence, coexistence and extinction in two species chemotaxis models on bounded heterogeneous environments. J. Dyn. Differ. Equations 31(4), 1839–1871 (2019)
Yurk, B.P., Cobbold, C.A.: Homogenization techniques for population dynamics in strongly heterogeneous landscapes. J. Biol. Dyn. 12(1), 171–193 (2018)
Dolak, Y., Hillen, T.: Cattaneo models for chemosensitive movement: numerical solution and pattern formation. J. Math. Biol. 46(5): 461–478 (2003)
Filbet, F., Laurencot, P., Perthame, B.: Derivation of hyperbolic models for chemosensitive movement. J. Math. Biol. 50(2), 189–207 (2005)
Tello, I.J., Wrzosek, D.: Predator–prey model with diffusion and indirect prey-taxis. Math. Models Methods Appl. Sci. 26(11), 2129–2162 (2016)
Li, U., Tao, Y.: Boundedness in a chemotaxis system with indirect signal production and generalized logistic source. Appl. Math. Lett. 77, 108–113 (2018)
Morgulis, A., Ilin, K.: Drift, stabilizing and destabilizing for a Keller-Siegel’s system with the short-wavelength external signal (2019). arXiv:1908.07075
Iooss, G., Joseph, D.D.: Elementary stability and bifurcation theory. Springer, Berlin (2012)
Arnold, V.I., Afrajmovich, V.S., Il’yashenko, Y.S., Shil’nikov, L.P.: Dynamical systems V: bifurcation theory and catastrophe theory. Springer, Berlin (2013)
Haragus, M., Iooss, G.: Local bifurcations, center manifolds, and normal forms in infinite-dimensional dynamical systems. Springer, Berlin (2010)
Yudovich, V.: The dynamics of vibrations in systems with constraints. Doklady Phys. 42, 322–325 (1997)
Vladimirov, V.: On vibrodynamics of pendulum and submerged solid. J. Math. Fluid Mech. 7(S3), S397–S412 (2005)
Vladimirov, V.: Two-Timing Hypothesis, Distinguished Limits, Drifts, and Pseudo-Diffusion for Oscillating Flows. Studies Appl. Math. 138(3), 269–293 (2017)
Yudovich, V.: Cycle-creating bifurcation from a family of equilibria of a dynamical system and its delay. J. Appl. Math. Mech. 62(1), 19–29 (1998)
Morshneva, I., Yudovich, V.: Bifurcation of cycles from equilibria of inversion-and rotation-symmetric dynamical systems. Sib. Math. J. 26(1), 97–104 (1985)
Allaire, G.: Shape optimization by the homogenization method. Springer, Berlin (2012)
Nirenberg, L.: A strong maximum principle for parabolic equations. Commun. Pure Appl. Math. 6(2), 167–177 (1953)
Landis, E.M.: Second order equations of elliptic and parabolic type. American Mathematical Society, New York (1997)
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Andrey Morgulis acknowledges the support from Southern Federal University.
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Morgulis, A., Ilin, K. (2021). Homogenization, Drift, Stabilization and Destabilization for the Patlak-Keller-Segel Systems Driven by the Indirect Taxis and the Short-Wavelength External Signals. In: Karapetyants, A.N., Kravchenko, V.V., Liflyand, E., Malonek, H.R. (eds) Operator Theory and Harmonic Analysis. OTHA 2020. Springer Proceedings in Mathematics & Statistics, vol 357. Springer, Cham. https://doi.org/10.1007/978-3-030-77493-6_25
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