Understanding the interplay between density dependent birth function and maturation time delay using a reaction-diffusion population model

We propose a new approach for proving uniqueness of semi-wavefronts in generally nonmonotone monostable reaction-diffusion equations with distributed delay. This allows to solve an open problem concerning the uniqueness of non-monotone (hence, slowly oscillating) semi-wavefronts to the KPP-Fisher equation with delay. Similarly, a broad family of the Mackey-Glass type diffusive equations is shown to possess a unique (up to translation) semi-wavefront for each admissible speed.

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“…A similar situation is also observed for another popular delayed model, the Mackey-Glass type diffusive equation [1,3,11,21,24,26,28,29,31,33,34]…”

Section: Introduction and Main Resultssupporting

confidence: 74%

A simple approach to the wave uniqueness problem

Solar

Trofimchuk

2019

Journal of Differential Equations

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“…In the next section, we apply Theorems 1.3 and 1.4 to models (3) and (4). In particular, we prove that condition (M) is fulfilled for these equations.…”

Section: Introduction and Main Resultsmentioning

confidence: 84%

“…Note that bistable equation (3) with the unimodal birth function f having only three fixed points, e 1 , e 2 , e 3 , is broadly used in the mathematical ecology for modelling systems exhibiting the Allee effect, cf. [4,Fig. 1d].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The above mentioned biological models show the importance of studying the existence of wavefronts for equation (1) with the reaction term g(u, v) which is not increasing in the second variable, but still has reasonably good (piece-wise monotone, with only two pieces of monotonicity) behavior with respect to v for each fixed u. We will include both models (3) and (4) in our general theory by considering two following alternative unimodality assumptions:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…On the other hand, the form of subtangency given in (U * ) seems to be more friendly for applications. For instance, it is easy to see that the reaction term in (3) satisfies (U * ) if f ′ (e 3 ) = min{f ′ (u), u ∈ [e 1 , e 3 ]} (note also that hypothesis (U) holds for equation (4) without additional restrictions on f ). Importantly, in Section 2 we show how a slightly weaker version of Theorem 1.4, Theorem 2.5, can be obtained without any kind of sub-tangency restriction at e 3 .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities

Trofimchuk

Volpert

2019

Nonlinearity

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We study the existence of monotone wavefronts for a general family of bistable reaction-diffusion equations with delayed reaction term g. Differently from previous works, we do not assume the monotonicity of g(u, v) with respect to the delayed variable v that does not allow to apply the comparison techniques. Thus our proof is based on a variant of the Hale-Lin functional-analytic approach to heteroclinic solutions of functional differential equations where Lyapunov-Schmidt reduction is done in appropriate weighted spaces of C 2 -smooth functions. This method requires a detailed analysis of associated linear differential Fredholm operators and their formal adjoints. For two different types of v−unimodal functions g(u, v), we prove the existence of a maximal continuous family of bistable monotone wavefronts.. Depending on the type of unimodality (equivalently, on the sign of the wave speed), two different scenarios can be observed for the bistable waves: 1) independently on the size of delay, each bistable wavefront is monotone; 2) wavefronts are monotone for moderate values of delays and can oscillate for large delays.AMS classification scheme numbers: 34K12, 35K57, 92D25

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Radially varying delay‐compensated distributed control of reaction‐diffusion PDEs on n‐ball under revolution symmetry conditions

Guan

2022

Intl J Robust & Nonlinear

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In this article, a distributed control design is developed via partial differential equation (PDE) backstepping for a class of reaction-diffusion PDEs on n-ball under revolution symmetry conditions in the presence of radial spatially varying actuator delay. The actuator delay is modeled by a two-dimensional (2D) first-order hyperbolic PDE with spatially varying boundary, resulting in a cascaded transport-diffusion PDE system. Two-step backstepping transformation is introduced for the control design. First, a new Volterra transformation for the n-ball under revolution symmetry is proposed that allows the kernel function to be solvable. Different from the kernel in the 1D domain, which is expressed by the classical Fourier series, the kernel of the n-ball under revolution symmetry conditions is based on the Fourier-Bessel series. The resulting Volterra transformation is an implicit one that contains the state of the target system on both sides of the definition. Based on the implicit transformation, we employ the successive approximations approach to derive the second-step transformation, namely, an explicit transformation, and arrive at a stable target system. The inverse transformation is also derived, to prove L 2 exponential stability of the closed-loop system.The stability analysis requires more technical skills that are not needed in solving the spatially varying delay compensation problem when the domain is 1D.Numerical simulation examples of 3D and 4D revolution-symmetrical systems illustrate the effectiveness of the controller.

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Understanding the interplay between density dependent birth function and maturation time delay using a reaction-diffusion population model

Cited by 26 publications

References 28 publications

A simple approach to the wave uniqueness problem

A simple approach to the wave uniqueness problem

Global continuation of monotone waves for bistable delayed equations with unimodal nonlinearities

Radially varying delay‐compensated distributed control of reaction‐diffusion PDEs on n‐ball under revolution symmetry conditions

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