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19 pages, 337 KB

Open AccessArticle

Prey-Taxis vs. An External Signal: Short-Wave Asymptotic and Stability Analysis

by Andrey Morgulis and Karrar H. Malal

Mathematics 2025, 13(2), 261; https://doi.org/10.3390/math13020261 - 14 Jan 2025

Cited by 1 | Viewed by 1458

We consider two models of the predator–prey community with prey-taxis. Both models take into account the capability of the predators to respond to prey density gradients and also to one more signal, the production of which occurs independently of the community state (such [...] Read more.

We consider two models of the predator–prey community with prey-taxis. Both models take into account the capability of the predators to respond to prey density gradients and also to one more signal, the production of which occurs independently of the community state (such a signal can be due to the spatiotemporal inhomogeneity of the environment arising for natural or artificial reasons). We call such a signal external. The models differ to one another through the description of their responses: the first one employs the Patlak–Keller–Segel law for both responses, and the second one employs Cattaneo’s model of heat transfer for both responses following to Dolak and Hillen. Assuming a short-wave external signal, we construct the complete asymptotic expansions of the short-wave solutions to both models. We use them to examine the effect of the short-wave signal on the formation of spatiotemporal patterns. We do so by comparing the stability of equilibria with no signal to that of the quasi-equilibria forced by the external signal. Such an approach refers back to Kapitza’s theory for an upside-down pendulum. The overall conclusion is that the external signal is likely not capable of creating the instability domain in the parametric space from nothing but it can substantially widen the one that is non-empty with no signal. Full article

(This article belongs to the Collection Theoretical and Mathematical Ecology)

15 pages, 273 KB

Open AccessArticle

Boundedness of Solutions for an Attraction–Repulsion Model with Indirect Signal Production

by Jie Wu and Yujie Huang

Mathematics 2024, 12(8), 1143; https://doi.org/10.3390/math12081143 - 10 Apr 2024

Cited by 15 | Viewed by 2276

Abstract

In this paper, we consider the following two-dimensional chemotaxis system of attraction–repulsion with indirect signal production [...] Read more.

In this paper, we consider the following two-dimensional chemotaxis system of attraction–repulsion with indirect signal production

\{\begin{matrix} 𝜕_{t} u = Δ u - \nabla \cdot (χ_{1} u \nabla v_{1}) + \nabla \cdot (χ_{2} u \nabla v_{2}), & x \in R^{2}, t > 0, \\ 0 = Δ v_{j} - λ_{j} v_{j} + w, & x \in R^{2}, t > 0, (j = 1, 2), \\ 𝜕_{t} w + δ w = u, & x \in R^{2}, t > 0, \\ u (0, x) = u_{0} (x), w (0, x) = w_{0} (x), & x \in R^{2}, \end{matrix}

where the parameters

χ_{i} \geq 0, λ_{i} > 0 (i = 1, 2)

and non-negative initial data

(u_{0} (x), w_{0} (x)) \in L^{1} (R^{2}) \cap L^{\infty} (R^{2})

. We prove the global bounded solution exists when the attraction is more dominant than the repulsion in the case of

χ_{1} \geq χ_{2}

. At the same time, we propose that when the radial solution satisfies

χ_{1} - χ_{2} \leq \frac{2 π δ}{∥ u_{0} ∥_{L^{1} (R^{2})} + {∥ w_{0} ∥}_{L^{1} (R^{2})}}

, the global solution is bounded. During the proof process, we found that adding indirect signals can constrict the blow-up of the global solution. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations, 2nd Edition)

13 pages, 484 KB

Open AccessFeature PaperArticle

Acoustic Wind in a Hyperbolic Predator—Prey System

by Andrey Morgulis

Mathematics 2023, 11(5), 1265; https://doi.org/10.3390/math11051265 - 6 Mar 2023

Cited by 2 | Viewed by 1970

Abstract

We address a hyperbolic model for prey-sensitive predators interacting with purely diffusive prey. We adopt the Cattaneo formulation for describing the predators’ transport. Given the hyperbolicity, the long-lived short-wave patterns occur for sufficiently weak prey diffusivities. The main result is that the non-linear [...] Read more.

We address a hyperbolic model for prey-sensitive predators interacting with purely diffusive prey. We adopt the Cattaneo formulation for describing the predators’ transport. Given the hyperbolicity, the long-lived short-wave patterns occur for sufficiently weak prey diffusivities. The main result is that the non-linear interplay of the short waves generically excites the slowly growing amplitude modulation for wide ranges of the model parameters. We have observed such a feature in the numerical experiments and support our conclusions with a short-wave asymptotic solution in the limit of vanishing prey diffusivity. Our reasoning relies on the so-called homogenized system that governs slow evolutions of the amplitudes of the short-wave parcels. It includes a term (called wind) which is absent in the original model and only comes from averaging over the short waves. It is the wind that (unlike any of the other terms!) is capable of exciting the instability and pumping the growth of solutions. There is quite a definite relationship between the predators’ transport coefficients to be held for getting rid of the wind. Interestingly, this relationship had been introduced in prior studies of small-scale mosaics in the spatial distributions of some real-life populations. Full article

(This article belongs to the Collection Theoretical and Mathematical Ecology)

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17 pages, 796 KB

Open AccessFeature PaperArticle

Waves in a Hyperbolic Predator–Prey System

by Andrey Morgulis

Axioms 2022, 11(5), 187; https://doi.org/10.3390/axioms11050187 - 20 Apr 2022

Cited by 4 | Viewed by 2706

Abstract

We address a hyperbolic predator–prey model, which we formulate with the use of the Cattaneo model for chemosensitive movement. We put a special focus on the case when the Cattaneo equation for the flux of species takes the form of conservation law—that is, [...] Read more.

We address a hyperbolic predator–prey model, which we formulate with the use of the Cattaneo model for chemosensitive movement. We put a special focus on the case when the Cattaneo equation for the flux of species takes the form of conservation law—that is, we assume a special relation between the diffusivity and sensitivity coefficients. Regarding this relation, there are pieces arguing for its relevance to some real-life populations, e.g., the copepods (Harpacticoida), in the biological literature (see the reference list). Thanks to the mentioned conservatism, we get exact solutions describing the travelling shock waves in some limited cases. Next, we employ the numerical analysis for continuing these waves to a wider parametric domain. As a result, we discover smooth solitary waves, which turn out to be quite sustainable with small and moderate initial perturbations. Nevertheless, the perturbations cause shedding of the predators from the main core of the wave, which can be treated as a settling mechanism. Besides, the localized perturbations make waves, colliding with the main core and demonstrating peculiar quasi-soliton phenomena sometimes resembling the leapfrog playing. An interesting side result is the onset of the migration waves due to the explosion of overpopulated cores. Full article

(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Physics)

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15 pages, 341 KB

Open AccessArticle

Multiscale Convergence of the Inverse Problem for Chemotaxis in the Bayesian Setting

by Kathrin Hellmuth, Christian Klingenberg, Qin Li and Min Tang

Computation 2021, 9(11), 119; https://doi.org/10.3390/computation9110119 - 11 Nov 2021

Cited by 5 | Viewed by 3649

Abstract

Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller–Segel equation and a chemotaxis kinetic equation. These two equations describe [...] Read more.

Chemotaxis describes the movement of an organism, such as single or multi-cellular organisms and bacteria, in response to a chemical stimulus. Two widely used models to describe the phenomenon are the celebrated Keller–Segel equation and a chemotaxis kinetic equation. These two equations describe the organism’s movement at the macro- and mesoscopic level, respectively, and are asymptotically equivalent in the parabolic regime. The way in which the organism responds to a chemical stimulus is embedded in the diffusion/advection coefficients of the Keller–Segel equation or the turning kernel of the chemotaxis kinetic equation. Experiments are conducted to measure the time dynamics of the organisms’ population level movement when reacting to certain stimulation. From this, one infers the chemotaxis response, which constitutes an inverse problem. In this paper, we discuss the relation between both the macro- and mesoscopic inverse problems, each of which is associated with two different forward models. The discussion is presented in the Bayesian framework, where the posterior distribution of the turning kernel of the organism population is sought. We prove the asymptotic equivalence of the two posterior distributions. Full article

(This article belongs to the Special Issue Inverse Problems with Partial Data)

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11 pages, 254 KB

Open AccessReview

A Review on the Qualitative Behavior of Solutions in Some Chemotaxis–Haptotaxis Models of Cancer Invasion

by Yifu Wang

Mathematics 2020, 8(9), 1464; https://doi.org/10.3390/math8091464 - 1 Sep 2020

Cited by 6 | Viewed by 2778

Abstract

Chemotaxis is an oriented movement of cells and organisms in response to chemical signals, and plays an important role in the life of many cells and microorganisms, such as the transport of embryonic cells to developing tissues and immune cells to infection sites. [...] Read more.

Chemotaxis is an oriented movement of cells and organisms in response to chemical signals, and plays an important role in the life of many cells and microorganisms, such as the transport of embryonic cells to developing tissues and immune cells to infection sites. Since the pioneering works of Keller and Segel, there has been a great deal of literature on the qualitative analysis of chemotaxis systems. As an important extension of the Keller–Segel system, a variety of chemotaxis–haptotaxis models have been proposed in order to gain a comprehensive understanding of the invasion–metastasis cascade. From a mathematical point of view, the rigorous analysis thereof is a nontrivial issue due to the fact that partial differential equations (PDEs) for the quantities on the macroscale are strongly coupled with ordinary differential equations (ODEs) modeling the subcellular events. It is the goal of this paper to describe recent results of some chemotaxis–haptotaxis models, inter alia macro cancer invasion models proposed by Chaplain et al., and multiscale cancer invasion models by Stinner et al., and also to introduce some open problems. Full article

(This article belongs to the Special Issue Modern Analysis and Partial Differential Equation)

15 pages, 5332 KB

Open AccessArticle

Spatiotemporal Pattern Formation in a Prey-Predator System: The Case Study of Short-Term Interactions Between Diatom Microalgae and Microcrustaceans

by Yuri V. Tyutyunov, Anna D. Zagrebneva and Andrey I. Azovsky

Mathematics 2020, 8(7), 1065; https://doi.org/10.3390/math8071065 - 1 Jul 2020

Cited by 21 | Viewed by 3063

Abstract

A simple mathematical model capable of reproducing formation of small-scale spatial structures in prey–predator system is presented. The migration activity of predators is assumed to be determined by the degree of their satiation. The hungrier individual predators migrate more frequently, randomly changing their [...] Read more.

A simple mathematical model capable of reproducing formation of small-scale spatial structures in prey–predator system is presented. The migration activity of predators is assumed to be determined by the degree of their satiation. The hungrier individual predators migrate more frequently, randomly changing their spatial position. It has previously been demonstrated that such an individual response to local feeding conditions leads to prey–taxis and emergence of complex spatiotemporal dynamics at population level, including periodic, quasi-periodic and chaotic regimes. The proposed taxis–diffusion–reaction model is applied to describe the trophic interactions in system consisting of benthic diatom microalgae and harpacticoid copepods. The analytical condition for the oscillatory instability of the homogeneous stationary state of species coexistence is given. The model parameters are identified on the basis of field observation data and knowledge on the species ecology in order to explain micro-scale spatial patterns of these organisms, which still remain obscure, and to reproduce in numerical simulations characteristic size and the expected lifetime of density patches. Full article

(This article belongs to the Special Issue Mathematical Modelling in Studying Spatial Aspects of Population Dynamics)

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12 pages, 317 KB

Open AccessFeature PaperArticle

Green’s Function of the Linearized Logarithmic Keller–Segel–Fisher/KPP System

by Jean Rugamba and Yanni Zeng

Math. Comput. Appl. 2018, 23(4), 56; https://doi.org/10.3390/mca23040056 - 3 Oct 2018

Cited by 3 | Viewed by 3672

Abstract

We consider a Keller–Segel type chemotaxis model with logarithmic sensitivity and logistic growth. The logarithmic singularity in the system is removed via the inverse Hopf–Cole transformation. We then linearize the system around a constant equilibrium state, and obtain a detailed, pointwise description of [...] Read more.

We consider a Keller–Segel type chemotaxis model with logarithmic sensitivity and logistic growth. The logarithmic singularity in the system is removed via the inverse Hopf–Cole transformation. We then linearize the system around a constant equilibrium state, and obtain a detailed, pointwise description of the Green’s function. The result provides a complete solution picture for the linear problem. It also helps to shed light on small solutions of the nonlinear system. Full article

(This article belongs to the Special Issue Selected Papers from the 2nd Global Conference on Applied Physics, Mathematics and Computing (APMC-18))

17 pages, 607 KB

Open AccessFeature PaperArticle

A (1 + 2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions. II

by Roman Cherniha and Maksym Didovych

Symmetry 2017, 9(1), 13; https://doi.org/10.3390/sym9010013 - 20 Jan 2017

Cited by 6 | Viewed by 5780

Abstract

A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry [...] Read more.

A simplified Keller–Segel model is studied by means of Lie symmetry based approaches. It is shown that a (1 + 2)-dimensional Keller–Segel type system, together with the correctly-specified boundary and/or initial conditions, is invariant with respect to infinite-dimensional Lie algebras. A Lie symmetry classification of the Cauchy problem depending on the initial profile form is presented. The Lie symmetries obtained are used for reduction of the Cauchy problem to that of (1 + 1)-dimensional. Exact solutions of some (1 + 1)-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1 + 1)-dimensional simplified Keller–Segel system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain a unique solution. The Lie symmetry classification of the (1 + 2)-dimensional Neumann problem for the simplified Keller–Segel system is derived. Because Lie symmetry of boundary-value problems depends essentially on geometry of the domain, which the problem is formulated for, all realistic (from applicability point of view) domains were examined. Reduction of the the Neumann problem on a strip is derived using the symmetries obtained. As a result, an exact solution of a nonlinear two-dimensional Neumann problem on a finite interval was found. Full article

(This article belongs to the Special Issue Lie and Conditional Symmetries and Their Applications for Solving Nonlinear Models)

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12 pages, 298 KB

Open AccessArticle

A (1+2)-Dimensional Simplified Keller–Segel Model: Lie Symmetry and Exact Solutions

by Maksym Didovych

Symmetry 2015, 7(3), 1463-1474; https://doi.org/10.3390/sym7031463 - 24 Aug 2015

Cited by 6 | Viewed by 4914

Abstract

This research is a natural continuation of the recent paper “Exact solutions of the simplified Keller–Segel model” (Commun Nonlinear Sci Numer Simulat 2013, 18, 2960–2971). It is shown that a (1+2)-dimensional Keller–Segel type system is invariant with respect infinite-dimensional Lie algebra. All possible [...] Read more.

This research is a natural continuation of the recent paper “Exact solutions of the simplified Keller–Segel model” (Commun Nonlinear Sci Numer Simulat 2013, 18, 2960–2971). It is shown that a (1+2)-dimensional Keller–Segel type system is invariant with respect infinite-dimensional Lie algebra. All possible maximal algebras of invariance of the Neumann boundary value problems based on the Keller–Segel system in question were found. Lie symmetry operators are used for constructing exact solutions of some boundary value problems. Moreover, it is proved that the boundary value problem for the (1+1)-dimensional Keller–Segel system with specific boundary conditions can be linearized and solved in an explicit form. Full article

(This article belongs to the Special Issue Lie Theory and Its Applications)

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15 pages, 1582 KB

Open AccessArticle

Analysis of the Keller–Segel Model with a Fractional Derivative without Singular Kernel

by Abdon Atangana and Badr Saad T. Alkahtani

Entropy 2015, 17(6), 4439-4453; https://doi.org/10.3390/e17064439 - 23 Jun 2015

Cited by 280 | Viewed by 10358

Abstract

Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence [...] Read more.

Using some investigations based on information theory, the model proposed by Keller and Segel was extended to the concept of fractional derivative using the derivative with fractional order without singular kernel recently proposed by Caputo and Fabrizio. We present in detail the existence of the coupled-solutions using the fixed-point theorem. A detailed analysis of the uniqueness of the coupled-solutions is also presented. Using an iterative approach, we derive special coupled-solutions of the modified system and we present some numerical simulations to see the effect of the fractional order. Full article

(This article belongs to the Special Issue Wavelets, Fractals and Information Theory I)

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