Spatial structures and generalized travelling waves for an integro-differential equation
Narcisa Apreutesei Nikolai Bessonov Vitaly Volpert Vitali Vougalter
Discrete & Continuous Dynamical Systems - B 2010, 13(3): 537-557 doi: 10.3934/dcdsb.2010.13.537
Some models in population dynamics with intra-specific competition lead to integro-differential equations where the integral term corresponds to nonlocal consumption of resources [8][9]. The principal difference of such equations in comparison with traditional reaction-diffusion equation is that homogeneous in space solutions can lose their stability resulting in emergence of spatial or spatio-temporal structures [4]. We study the existence and global bifurcations of such structures. In the case of unbounded domains, transition between stationary solutions can be observed resulting in propagation of generalized travelling waves (GTW). GTWs are introduced in [18] for reaction-diffusion systems as global in time propagating solutions. In this work their existence and properties are studied for the integro-differential equation. Similar to the reaction-diffusion equation in the monostable case, we prove the existence of generalized travelling waves for all values of the speed greater or equal to the minimal one. We illustrate these results by numerical simulations in one and two space dimensions and observe a variety of structures of GTWs.
keywords: Integro-differential equations generalized travelling waves. population dynamics
Reaction-diffusion waves with nonlinear boundary conditions
Narcisa Apreutesei Vitaly Volpert
Networks & Heterogeneous Media 2013, 8(1): 23-35 doi: 10.3934/nhm.2013.8.23
A reaction-diffusion equation with nonlinear boundary condition is considered in a two-dimensional infinite strip. Existence of waves in the bistable case is proved by the Leray-Schauder method.
keywords: existence Leray-Schauder method. nonlinear boundary conditions Travelling waves
On the solvability conditions for the diffusion equation with convection terms
Vitali Vougalter Vitaly Volpert
Communications on Pure & Applied Analysis 2012, 11(1): 365-373 doi: 10.3934/cpaa.2012.11.365
A linear second order elliptic equation describing heat or mass diffusion and convection on a given velocity field is considered in $R^3$. The corresponding operator $L$ may not satisfy the Fredholm property. In this case, solvability conditions for the equation $L u = f$ are not known. In this work, we derive solvability conditions in $H^2(R^3)$ for the non self-adjoint problem by relating it to a self-adjoint Schrödinger type operator, for which solvability conditions are obtained in our previous work [13].
keywords: Solvability conditions porous medium continuous spectrum. adjoint operator
Travelling waves for integro-differential equations in population dynamics
Narcisa Apreutesei Arnaud Ducrot Vitaly Volpert
Discrete & Continuous Dynamical Systems - B 2009, 11(3): 541-561 doi: 10.3934/dcdsb.2009.11.541
The paper is devoted to integro-differential equations arising in population dynamics. The integral term describes the nonlocal consumption of resources. We study the Fredholm property of the corresponding linear operators and use it to prove the existence of travelling waves when the support of the integral is sufficiently small. In this case, the integro-differential operator is close to the differential operator and we can use the implicit function theorem. We carry out numerical simulations in order to study the case where the support of the integral is not small. We observe various regimes of wave propagation. Some of them, in particular periodic waves do not exist for the usual reaction-diffusion equation.
keywords: travelling waves. Integro-differential equation Fredholm property

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