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Non-local reaction-diffusion equations modelling predator-prey coevolution
Abstract
In this paper we examine a prey-predator system with a characteristic of the predator subject to mutation. The ultimate equilibrium of the system is found by Maynard-Smith et al. by the so called ESS (Evolutionary Stable Strategy). Using a system of reaction-diffusion equations with non local terms, we conclude the ESS result for the diffusion coefficient tending to zero, without resorting to any optimization criterion.
... The nonlocal Fisher-KPP equation (3) is called the "competitive Lotka-Volterra model" [334][335][336]. It is a simple nonlocal model with many applications, e.g., nonlocal intra-specific competition in the predatorprey model [88, 90, 92, 96-98, 101, 131, 337], global consumption of resources [338], fear effects of prey species [339], cooperative behaviour [340]. As mentioned earlier, the homogeneous solution u = 1 can be unstable due to nonlocal interaction. ...
Mathematical modelling is one of the fundamental techniques for understanding biophysical mechanisms in developmental biology. It helps researchers to analyze complex physiological processes and connects like a bridge between theoretical and experimental observations. Various groups of mathematical models have been studied to analyze these processes, and the nonlocal models are one of them. Nonlocality is important in realistic mathematical models of physical and biological systems when local models fail to capture the essential dynamics and interactions that occur over a range of distances (e.g., cell-cell, cell-tissue adhesions, neural networks, the spread of diseases, intra-specific competition, nanobeams, etc.). This review illustrates different nonlocal mathematical models applied to biology and life sciences. The major focus has been given to sources, developments, and applications of such models. Among other things, a systematic discussion has been provided for the conditions of pattern formations in biological systems of population dynamics. Special attention has also been given to nonlocal interactions on networks, network coupling and integration, including brain dynamics models that provide an important tool to understand neurodegenerative diseases better. In addition, we have discussed nonlocal modelling approaches for cancer stem cells and tumor cells that are widely applied in the cell migration processes, growth, and avascular tumors in any organ. Furthermore, the discussed nonlocal continuum models can go sufficiently smaller scales, including nanotechnology, where classical local models often fail to capture the complexities of nanoscale interactions, applied to build biosensors to sense biomaterial and its concentration. Piezoelectric and other smart materials are among them, and these devices are becoming increasingly important in the digital and physical world that is intrinsically interconnected with biological systems. Additionally, we have reviewed a nonlocal theory of peridynamics, which deals with continuous and discrete media and applies to model the relationship between fracture and healing in cortical bone, tissue growth and shrinkage, and other areas increasingly important in biomedical and bioengineering applications. Finally, we provided a comprehensive summary of emerging trends and highlighted future directions in this rapidly expanding field.
... However, these models do not incorporate eco-evolutionary dynamics. Integro-differential equation models which do incorporate eco-evolutionary dynamics have provided unique insights into evolutionary branching [24], the interaction between stromal cells and stem cells [25], and the co-evolution of predators and prey [26]. While these models specifically involve mutational processes, they focus on stationary solutions (the evolutionary stable strategies in the language of adaptive dynamics) they miss the transient dynamics which may be of interest in applications, especially if realistic parameters would require a longer time to converge than the time-frame of interest. ...
The coupling between evolutionary and ecological changes (eco-evolutionary dynamics) has been shown to be relevant among diverse species, and is also of interest outside of ecology, i.e. in cancer evolution. These dynamics play an important role in determining survival in response to climate change, motivating the need for mathematical models to capture this often complex interplay. Population genetics models incorporating eco-evolutionary dynamics often sacrifice analytical tractability to model the complexity of real systems, and do not explicitly consider the effect of population heterogeneity. In order to allow for population heterogeneity, transient, and long-term dynamics, while retaining tractability, we generalise a moment-based method applicable in the regime of small segregational variance to the case of time-dependent mortality and birth. These results are applied to a predator-prey model, where ecological parameters such as the contact rate between species are trait-structured. The trait-distribution of the prey species is shown to be approximately Gaussian with constant variance centered on the mean trait, which is asymptotically governed by an autonomous ODE. In this way, we make explicit the impact of eco-evolutionary dynamics on the transient behaviour and long-term fate of the prey species.
... The blow-up and global phenomena described by (A) have been strongly considered with respects to suitable initial data and different boundary conditions. Equation (A) with local/nonlocal nonlinearity or coupling of local and nonlocal nonlinearity has been studied in the interesting papers [28, 1-3, 11, 12, 36, 37,33,5], which have been applied in the study of population dynamics [13], biological evolution [8], combustion theory [15], and others. The pioneering works for blow-up solutions with nonlocal reaction may be due to Pao [25], and Souplet [28]. ...
In this paper, we are concerned with the characterization of the blow-up and global solutions for free boundary parabolic equation with competing nonlocal nonlinearity and absorption
where , p=0 or and are given constants. This study is motivated from the works [Abdelhedi and Zaag: J. Differential Equations 272, 1–45, (2021); Souplet: SIAM J. Math. Anal. 29, 1301–1334, (1998); Zhou and Lin: J. Funct. Anal. 262, 3409–3429, (2012)] arisen from the investigation of many physical and biological phenomena such as population dynamics, combustion theory, phase separation in binary mixtures, theory of nuclear reactor dynamics... We first prove the local existence, uniqueness and stability of solution thanks to the “extension trick" introduced in [Du et al.: Math. Ann. 386(3-4), 2061–2106, (2023); Wang and Du: Discrete Contin. Dyn. Syst. Ser. B 26(4), 2201–2238, (2021)]. Second, by improving the comparison principle used in [Souplet: SIAM J. Math. Anal. 29, 1301–1334, (1998); Zhou and Lin: J. Funct. Anal. 262, 3409–3429, (2012)], we find a sharp criterion characterizing the blow-up and global solutions of (1) in term of power coefficients and initial data. We further show that there exists a threshold for the initial data that determines whether blow-up, global fast, or global slow solutions occur and find an upper bound for the existence time of blow-up solutions in two different cases and . Our proofs are mainly based on the comparison principle by improving several techniques in previous works, combined new idea to handle differential inequalities and unified local existence theory for nonlocal semilinear parabolic equations.
... The gradient model (1.1) is often referred to as a viscous Hamilton-Jacobi equation and appears in many natural phenomena such as explosion model, compressible reactant gas model, population dynamics theories and some biological species with a human-controlled distribution model, see [1,3,6] and references therein. The formulation (1.1) also describes the evolution of some biological population u on a certain occupied region and whose growth is governed by the law of f , see [7]. ...
... For instance, when a = 1, such system appears in population dynamics, where y(t, x) represents the density of the species at position x and time t, while the reaction term 1 0 K(·, ·, τ )y(·, τ ) dτ is considered as the rate of reproduction. This integral term is a way to express that the evolution of the species in a point of space depends on the total amount of the species (see for instance [10,23,32]). ...
We consider two degenerate heat equations with a nonlocal space term, studying, in particular, their null controllability property. To this aim, we first consider the associated nonhomogeneous degenerate heat equations: we study their well posedness, the Carleman estimates for the associated adjoint problems and, finally, the null controllability. Then, as a consequence, using the Kakutani's fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.
... For instance, when a = 1, such system appears in population dynamics, where y(t, x) represents the density of the species at position x and time t, while the reaction term 1 0 K(·, ·, τ )y(·, τ ) dτ is considered as the rate of reproduction. This integral term is a way to express that the evolution of the species in a point of space depends on the total amount of the species (see for instance [9,23,32]). ...
We consider two degenerate heat equations with a nonlocal space term, studying, in particular, their null controllability property. To this aim, we first consider the associated nonhomogeneous degenerate heat equations: we study their well posedness, the Carleman estimates for the associated adjoint problems and, finally, the null controllability. Then, as a consequence, using the Kakutani's fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.
This paper is concerned with the blow-up phenomena and global existence of a fractional nonlinear reaction-diffusion equation with a non-local source term. Under sufficient conditions on the weight function a(x) and when the initial data is small enough, the global existence of solutions is proved using the comparison principle. We establish a finite time blow-up of the solution with large initial data by converting the fractional PDE into a simple ordinary differential inequality using the differential inequality technique. Moreover, by solving the obtained ordinary differential inequality, an upper bound of the blow-up time is also deduced.
This paper presents several properties associated with the two-variable extension of the Chebyshev matrix polynomials of the second kind. In particular, we establish a three-term recurrence relation for these two-variable matrix polynomials and show that these two-variable matrix polynomials satisfy some second-order matrix differential equations. We derive their hypergeometric matrix representation and an expansion formula which links these generalized Chebyshev matrix polynomials with the Hermite matrix polynomials and the Laguerre matrix polynomials. We also drive their Volterra integral equation.
In this paper, homotopy analysis method (HAM) is used to obtain the analytic solution for fragmentation population balance equation. Different sample problems are solved using HAM and their series solution is obtained. A detailed analysis of the series solution and the region of convergence of the solution is also studied. It is observed that the convergence region of the series solution can be adjusted with the help of certain parameters involved in HAM.KeywordsHomotopy analysis methodPopulation balance equationAnalytic approximationsFragmentationConvergence
In this paper, we investigate a reaction–diffusion equation u t − d u x x = u ( a + b u p − 1 ∫ g ( t ) h ( t ) u q d x ) with double free boundaries. We study blowup phenomena and asymptotic behavior of time-global solutions. For u 0 ( x ) = σ ϕ ( x ) , when a ≥ λ 1 , σ > 0 , if h 0 ≥ π 2 d a , then u will blow up in finite time. Meanwhile, we also prove when T ∗ < + ∞ , the solution must blow up in finite time. On the other hand, we discuss the vanishing of solutions. We prove that vanishing will occur as long as the initial datum are small sufficiently.
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