arXiv subjects

Karrar Malal

Publications and source records attributed to Karrar Malal.

3 recordsLinked to original sources

arXiv2026

Short-wave signal versus indirect prey-taxis

We address a short-wave asymptotic for one class of quasi-linear second-order PDE systems involving the cross-diffusion described by the so-called Patlak-Keller-Segel law. It is common to employ these equations for modeling the predator-prey community with the prey-taxis that means the interactions of two species of particles or cells or anything else through which the species called "predators" is capable of moving directionally while searching for the other species called "prey." However, we suppose the predators to be sensitive not to the prey density but to a driving signal produced by the prey. Additionally, the production of the driving signal is assumed to be sensitive to the intensity of an external field, which is independent from the community state. This is what we call the external signal. It can be due to the spatiotemporal inhomogeneity of the environment arising from natural or artificial reasons. We assume that the external signal takes a general short-wave form and construct a complete asymptotic expansion for the short-wave solutions with no restrictions on the spatial dimension or kinetics of inter- or intra-specific reactions. Further, we apply the short wave asymptotic to studying the stability or instability induced by the external signal following Kapitza's theory for the upside-down pendulum. Applying the general results to some special classes of external signals, we get examples of suppressing the taxical transport, examples of robustness of the species equilibrium to the signal up to a very strong stabilization or, oppositely, destabilization and somewhat like blurring the borderline in the parametric space between the areas of stability and instability of this equilibrium. These results contribute to filling the gap in the literature, since the theory and techniques for the asymptotic integration of systems described above represent a weakly charted area.

math.AP↗

arXiv2025

Prey-taxis VS a Shortwave External Signal in Multiple Dimensions

We consider a model of the predator--prey community with prey-taxis. By that we mean the capability of the predators to get moving in a certain direction on the macroscopic level in response to the prey density gradients. Additionally, we suppose the same kind of sensitivity with respect to one more signal, called external, the production of which goes on independently of the community state. Such a signal can be due to the spatiotemporal inhomogeneity of the environment that results from the natural or artificial reasons. The model employs the Patlak--Keller--Segel law for responses to both ones. We assume that the external signal takes a general short-wave form, and we construct the complete asymptotic expansions of the short-wave solutions. This result generalizes the prior one by Morgulis \& Malal (2025) in two respects. First, we have addressed the case of multiple dimensions. Second, we have got rid of assuming the signal and corresponding solutions to take the form of a traveling wave, that makes our result novel even in one dimension. Further, we apply the short wave asymptotic to studying the stability or instability imposed by the external signal following Kapitza' theory for upside-down pendulum.

math.AP↗

arXiv2024

Prey-taxis vs an external signal: short-wave asymptotic and stability analysis

We consider two models of predator-prey community with prey-taxis, one relies on Patlak-Keller-Segel law (Lee et al, 2009), the other one employs the Cattaneo model of heat transfer following Dolak and Hillen (2003). Thus, the former one uses the prey density gradient for directing the predators flux, and the latter one -- for directing the vector density of sources of the predators flux. We assume the predators to be also capable of responding to an external signal in the same manner. Additionally, we assume that some scaling makes the dimensionless prey diffusivity and the wavelength of the external signal small quantities of the same order. In the case of Cattaneo's model, we additionally assume the resistivity to varying the predators flux to be as high as the reciprocal of the prey diffusivity. With these assumptions we construct the complete asymptotic expansion of the short-wave solution. We use this asymptotic 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, by which we mean the simplest patterns directly forced by the external signal. Historically, such an approach goes back to Kapitza's theory for upside-down pendulum, and it had been already applied to examining other predator-prey systems and/or other limits (Morgulis and Ilin, 2020, Morgulis, 2023). This time 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. The details, however, essentially depend on the speed at which the external signal propagates and on the system kinetics, at least when the signal amplitude is small

q-bio.PE↗