Starvation driven diffusion as a survival strategy of biological organisms

Fractionation by Spatially Heterogeneous Diffusion: Experiments and the Two-Component Random Walk Model

The fundamental question regarding the fractionation phenomenon is whether diffusion alone is responsible for it or whether an additional advection dynamic is involved. We studied the fractionation by diffusion of particles in spatially heterogeneous environments. By experimentally observing the time-sequential fractionation patterns of dye particles diffusing across a solid-solid interface of varying polyacrylamide gel densities, we found that the two-component diffusion model accurately captures the observed fractionation dynamics. In contrast, single-component diffusion models by Fick, Wereide, and Chapman do not. Our results indicate that diffusion alone can explain the fractionation phenomenon and that additional advection dynamics are not involved. The underlying physics in the fractionation phenomenon is discussed by using a two-component random walk model.

Biological invasion with a porous medium type diffusion in a heterogeneous space

The knowledge of traveling wave solutions is the main tool in the study of wave propagation. However, in a spatially heterogeneous environment, traveling wave solutions do not exist, and a different approach is needed. In this paper, we study the generation and the propagation of hyperbolic scale singular limits of a KPP-type reaction-diffusion equation when the carrying capacity is spatially heterogeneous and the diffusion is of a porous medium equation type. We show that the interface propagation speed varies according to the carrying capacity.

Genomic and single-cell analyses reveal genetic signatures of swimming pattern and diapause strategy in jellyfish

Jellyfish exhibit innovative swimming patterns that contribute to exploring the origins of animal locomotion. However, the genetic and cellular basis of these patterns remains unclear. Herein, we generated chromosome-level genome assemblies of two jellyfish species, Turritopsis rubra and Aurelia coerulea, which exhibit straight and free-swimming patterns, respectively. We observe positive selection of numerous genes involved in statolith formation, hair cell ciliogenesis, ciliary motility, and motor neuron function. The lineage-specific absence of otolith morphogenesis- and ciliary movement-related genes in T. rubra may be associated with homeostatic structural statocyst loss and straight swimming pattern. Notably, single-cell transcriptomic analyses covering key developmental stages reveal the enrichment of diapause-related genes in the cyst during reverse development, suggesting that the sustained diapause state favours the development of new polyps under favourable conditions. This study highlights the complex relationship between genetics, locomotion patterns and survival strategies in jellyfish, thereby providing valuable insights into the evolutionary lineages of movement and adaptation in the animal kingdom.

Predation-induced dispersal toward fitness for predator invasion in predator-prey models

In this paper, we consider a predator-prey model with nonuniform predator dispersal, called predation-induced dispersal (PID), which represents predator motility depending on the maximal predation rate and the predator death rate in a spatially heterogeneous region. We study the local stability of the semitrivial steady state when predators are absent for models with PID and linear dispersal. We then investigate the local/global bifurcation from the semitrivial steady state of these models. Finally, we compare the results of the model with PID to the results of the model with linear dispersal. We conclude that the nonuniform dispersal of predators obeying PID increases fitness for predator invasion when rare; thus, predators with PID can invade a region with an increased probability even in cases wherein predators dispersed linearly cannot invade a certain region. Based on the results, we provide an ecological interpretation with the simulations.

Modelling the interaction of invasive-invaded species based on the general Bramson dynamics and with a density dependant diffusion and advection

The main goal of the presented study is to introduce a model of a pairwise invasion interaction with a nonlinear diffusion and advection. The new equation is based on the further general works introduced by Bramson (1988) to describe the invasive-invaded dynamics. This type of model is made particular with a density dependent diffusion along with an advection term. The new resulting model is then analyzed to explore the regularity, existence and uniqueness of solutions. It is well known that density dependent diffusion operators induce a propagating front with finite speed for compactly supported functions. Based on this, we introduce an analytical approach to determine the evolution of such a propagating front in the invasion dynamics. Afterward, we study the problem with travelling wave profiles and a numerical assessment. As a main finding to remark: When both species propagate with significantly different travelling wave speeds, the interaction becomes unstable, while when the species propagate with similar low speeds, the interaction stabilizes.

Open problems in PDE models for knowledge-based animal movement via nonlocal perception and cognitive mapping

The inclusion of cognitive processes, such as perception, learning and memory, are inevitable in mechanistic animal movement modelling. Cognition is the unique feature that distinguishes animal movement from mere particle movement in chemistry or physics. Hence, it is essential to incorporate such knowledge-based processes into animal movement models. Here, we summarize popular deterministic mathematical models derived from first principles that begin to incorporate such influences on movement behaviour mechanisms. Most generally, these models take the form of nonlocal reaction-diffusion-advection equations, where the nonlocality may appear in the spatial domain, the temporal domain, or both. Mathematical rules of thumb are provided to judge the model rationality, to aid in model development or interpretation, and to streamline an understanding of the range of difficulty in possible model conceptions. To emphasize the importance of biological conclusions drawn from these models, we briefly present available mathematical techniques and introduce some existing "measures of success" to compare and contrast the possible predictions and outcomes. Throughout the review, we propose a large number of open problems relevant to this relatively new area, ranging from precise technical mathematical challenges, to more broad conceptual challenges at the cross-section between mathematics and ecology. This review paper is expected to act as a synthesis of existing efforts while also pushing the boundaries of current modelling perspectives to better understand the influence of cognitive movement mechanisms on movement behaviours and space use outcomes.

Population dynamics with resource-dependent dispersal: single- and two-species models

In this paper, we consider the population models with resource-dependent dispersal for single-species and two-species with competition. For the single-species model, it is well-known that the total population supported by the environment is always greater than the environmental carrying capacity if the dispersal is simply random diffusion. However, we find that the total population supported can be equal or smaller than the environmental carrying capacity when the dispersal depends on the resource distribution. This analytical finding not only well agrees with the yeast experiment observation of Zhang et al. (Ecol Lett 20(9):1118-1128, 2017), but also indicates that resource-dependent dispersal may produce different outcomes compared to the random dispersal. For the two-species competition model, when two competing species use the same dispersal strategy up to a multiplicative constant (i.e. their dispersal strategies are proportional) or both diffusion coefficients are large, we give a classification of global dynamics. We also show, along with numerical simulations, that if the dispersal strategies are resource-dependent, even one species has slower diffusion, coexistence is possible though competitive exclusion may occur under different conditions. This is distinct from the prominent result that with random dispersal the slower diffuser always wipes out its fast competitor. Our analytical results, supported by the numerical simulations, show that the resource-dependent dispersal strategy has profound impact on the population dynamics and evolutionary processes.

On the Fitness of Predators with Prey-Induced Dispersal in a Habitat with Spatial Heterogeneity

This study considers a situation in which a predator can change its dispersal rate according to its satisfaction with foraging prey in a predator-prey interaction. However, since it is impossible to accurately determine the magnitude of the density of prey that is favorable to a predator's survival in an area, the predator determines the movement rate through inaccurate judgment. In this situation, we investigate the effect of the predator's decision about its movement on fitness. To achieve our goal, we consider a predator-prey model with nonuniform predator dispersal, called prey-induced dispersal (PYID), in which the spread of predators is small when the prey density is larger than a certain value, and when the prey density is smaller than a particular value, a large spread of predators occurs. To understand how PYID affects the dynamics and coexistence of the system in a spatially heterogeneous region, we examine a model with Holling-type II functional responses under no-flux boundary conditions wherein the predators move according to the PYID. We study the local stability of the semitrivial solution of models with PYID and linear dispersal where the predator is absent. Furthermore, we investigate the local/global bifurcation from the semitrivial solution of models with two different dispersals. We conclude that in most cases, nonuniform dispersal of predators following PYID promotes predator fitness; however, there is a case in which PYID does not increase predator fitness. If a predator's satisfaction degree regarding the prey density is higher than a certain level, there may exist a case that is not beneficial for predators in terms of their fitness. However, if the satisfaction level of predators regarding prey density is relatively low, predators following PYID will take advantage of fitness. More precisely, if predators are dissatisfied with the amount of prey in a region and move quickly, even for abundant prey density, they may not benefit from PYID. Meanwhile, if predators change their motility when they are appropriately satisfied with the amount of prey, they will obtain a survival advantage. We obtain the results by analyzing an eigenvalue problem at the semitrivial solution from the linearized operators derived from the models.

Regularity, Asymptotic Solutions and Travelling Waves Analysis in a Porous Medium System to Model the Interaction between Invasive and Invaded Species

This work provides an analytical approach to characterize and determine solutions to a porous medium system of equations with views in applications to invasive-invaded biological dynamics. Firstly, the existence and uniqueness of solutions are proved. Afterwards, profiles of solutions are obtained making use of the self-similar structure that permits showing the existence of a diffusive front. The solutions are then studied within the Travelling Waves (TW) domain showing the existence of potential and exponential profiles in the stable connection that converges to the stationary solutions in which the invasive species predominates. The TW profiles are shown to exist based on the geometry perturbation theory together with an analytical-topological argument in the phase plane. The finding of an exponential decaying rate (related with the advection and diffusion parameters) in the invaded species TW is not trivial in the nonlinear diffusion case and reflects the existence of a TW trajectory governed by the invaded species runaway (in the direction of the advection) and the diffusion (acting in a finite speed front or support).

Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit

We rigorously prove the passage from a Lotka-Volterra reaction-diffusion system towards a cross-diffusion system at the fast reaction limit. The system models a competition of two species, where one species has a more diverse diet than the other. The resulting limit gives a cross-diffusion system of a starvation driven type. We investigate the linear stability of homogeneous equilibria of those systems and rule out the possibility of cross-diffusion induced instability (Turing instability). Numerical simulations are included which are compatible with the theoretical results.

Invasive-invaded system of non-Lipschitz porous medium equations with advection

This work provides analytical results towards applications in the field of invasive-invaded systems modeled with nonlinear diffusion and with advection. The results focus on showing regularity, existence and uniqueness of weak solutions using the condition of a nonlinear slightly positive parabolic operator and the reaction-absorption monotone properties. The coupling in the reaction-absorption terms, that characterizes the species interaction, impedes the formulation of a global comparison principle that is shown to exist locally. Additionally, this work provides analytical solutions obtained as selfsimilar minimal and maximal profiles. A propagating diffusive front is shown to exist until the invaded specie notes the existence of the invasive. When the desertion of the invaded starts, the diffusive front vanishes globally and the nonlinear diffusion concentrates only on the propagating tail which exhibits finite speed. Finally, the invaded specie is shown to exhibit an exponential decay along a characteristic curve. Such exponential decay is not trivial in the nonlinear diffusion case and confirms that the invasive continues to feed on the invaded during the desertion.

Intraguild predation with evolutionary dispersal in a spatially heterogeneous environment

In many cases, the motility of species in a certain region can depend on the conditions of the local habitat, such as the availability of food and other resources for survival. For example, if resources are insufficient, the motility rate of a species is high, as they move in search of food. In this paper, we present intraguild predation (IGP) models with a nonuniform random dispersal, called starvation-driven diffusion, which is affected by the local conditions of habitats in heterogeneous environments. We consider a Lotka-Volterra-type model incorporating such dispersals, to understand how a nonuniform random dispersal affects the fitness of each species in a heterogeneous region. Our conclusion is that a nonuniform dispersal increases the fitness of species in a spatially heterogeneous environment. The results are obtained through an eigenvalue analysis of the semi-trivial steady state solutions for the linearized operator derived from the model with nonuniform random diffusion on IGPrey and IGPredator, respectively. Finally, a simulation and its biological interpretations are presented based on our results.

Dispersal towards food: the singular limit of an Allen-Cahn equation

The effect of dispersal under heterogeneous environment is studied in terms of the singular limit of an Allen-Cahn equation. Since biological organisms often slow down their dispersal if food is abundant, a food metric diffusion is taken to include such a phenomenon. The migration effect of the problem is approximated by a mean curvature flow after taking the singular limit which now includes an advection term produced by the spatial heterogeneity of food distribution. It is shown that the interface moves towards a local maximum of the food distribution. In other words, the dispersal taken in the paper is not a trivialization process anymore, but an aggregation one towards food.

A Discrete Velocity Kinetic Model with Food Metric: Chemotaxis Traveling Waves

We introduce a mesoscopic scale chemotaxis model for traveling wave phenomena which is induced by food metric. The organisms of this simplified kinetic model have two discrete velocity modes, [Formula: see text] and a constant tumbling rate. The main feature of the model is that the speed of organisms is constant [Formula: see text] with respect to the food metric, not the Euclidean metric. The uniqueness and the existence of the traveling wave solution of the model are obtained. Unlike the classical logarithmic model case there exist traveling waves under super-linear consumption rates and infinite population pulse-type traveling waves are obtained. Numerical simulations are also provided.

Evolution of Dispersal with Starvation Measure and Coexistence

Many biological species increase their dispersal rate if starvation starts. To model such a behavior, we need to understand how organisms measure starvation and response to it. In this paper, we compare three different ways of measuring starvation by applying them to starvation-driven diffusion. The evolutional selection and coexistence of such starvation measures are studied within the context of Lotka-Volterra-type competition model of two species. We will see that, if species have different starvation measures and different motility functions, both the coexistence and selection are possible.

Evolution of dispersal in closed advective environments

We study a two-species competition model in a closed advective environment, where individuals are exposed to unidirectional flow (advection) but no individuals are lost through the boundary. The two species have the same growth and advection rates but different random dispersal rates. The linear stability analysis of the semi-trivial steady state suggests that, in contrast to the case without advection, slow dispersal is generally selected against in closed advective environments. We investigate the invasion exponent for various types of resource functions, and our analysis suggests that there might exist some intermediate dispersal rate that will be selected. When the diffusion and advection rates are small and comparable, we determine criteria for the existence and multiplicity of singular strategies and evolutionarily stable strategies. We further show that every singular strategy is convergent stable.

Bacterial chemotaxis without gradient-sensing

Chemotaxis models are based on spatial or temporal gradient measurements by individual organisms. The key contribution of Keller and Segel (J Theor Biol 30:225-234, 1971a; J Theor Biol 30:235-248, 1971b) is showing that erratic measurements of individuals may result in an accurate chemotaxis phenomenon as a group. In this paper we provide another option to understand chemotactic behavior when individuals do not sense the gradient of chemical concentration by any means. We show that, if individuals increase their dispersal rate to find food when there is not enough food, an accurate chemotactic behavior may be obtained without sensing the gradient. Such a dispersal has been suggested by Cho and Kim (Bull Math Biol 75:845-870, 2013) and was called starvation driven diffusion. This model is surprisingly similar to the original Keller-Segel model. A comprehensive picture of traveling bands and fronts is provided.

Evolution of dispersal toward fitness

It is widely believed that the slowest dispersal strategy is selected in the evolutional if the environment is temporally invariant but spatially heterogeneous. Authors claim in this paper that this belief is true only if random dispersals with constant motility are considered. However, if a dispersal strategy with fitness property is included, the size of the dispersal is not such a crucial factor anymore. Recently, a starvation driven diffusion has been introduced by Cho and Kim (Bull. Math. Biol., 2013), which is a random dispersal strategy with a motility increase on starvation. The authors show that such a dispersal strategy has fitness property and that the evolutional selection favors fitness but not simply slowness. Such a conclusion is obtained from a stability analysis of a competition system between two phenotypes with different dispersal strategies of linear and starvation driven diffusions.

Global asymptotic stability and the ideal free distribution in a starvation driven diffusion

We study a logistic model with a nonlinear random diffusion in a Fokker-Planck type law, but not in Fick's law. In the model individuals are assumed to increase their motility if they starve. Any directional information to resource is not assumed in this starvation driven diffusion and individuals disperse in a random walk style strategy. However, the non-uniformity in the motility produces an advection toward surplus resource. Several basic properties of the model are obtained including the global asymptotic stability and the acquisition of the ideal free distribution.