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This work investigates the complex dynamics of a discrete-time predator–prey system with a nonlinear Allee effect. We obtain the discrete system using the piecewise constant argument method. The piecewise argument approach produces a more dynamically consistent discrete system than other numerical techniques for discretization. We investigate the presence and stability of fixed points. Furthermore, we have demonstrated that the system undergoes Neimark–Sacker bifurcation at the positive fixed point by utilizing the Allee effect constant as the bifurcation parameter. To reduce bifurcation and chaos, we use feedback and hybrid control strategies. Our numerical simulations demonstrate the importance of the Allee effect in determining the system's behavior. The findings indicate that an adequate Allee effect might improve social connections and cooperation across populations. However, a significant Allee effect on prey can destabilize the positive fixed point, thus resulting in the extinction of predator and prey populations. [ABSTRACT FROM AUTHOR]

The harvesting of population has a dominant influence in balancing the ecosystem. In this manuscript, the impact of harvesting in addition to competition, and memory effect on a prey-predator interaction following the Lotka-Volterra model is studied. The mathematical validation is provided by proofing that all solutions of the model are always exist, non-negative, and bounded. Obeying Matignon condition, Lyapunov function, and generalized LaSalle invariance principle, the local and global stability are investigated. To complete the analytical results, some numerical simulations are given to show the occurrence of forward bifurcation and the impact of the memory index. All results state that three possible circumstances may occur namely the extinction of both populations, the prey-only population, and the co-existence of both populations.

The current study is based on a predator-prey model with infection that affects only predator species. Predators are divided into two categories such as the susceptible predator and the infected predator, which are feeding on prey species. The Takagi-Sugeno (T-S) based fuzzy impulsive control model was used to explore the stability of the Lotka-Volterra predator-prey system. Numerical simulation provides global stability and the fuzzy solution.

The aim of this paper is to examine the structure of genetic algebras associated with permuted Lotka-Volterra operators on a finite-dimensional simplex. Furthermore, three dimensional permuted Lotka-Voltarra algebras are considered and the description of their derivations are given as well. [ABSTRACT FROM AUTHOR]

In the Netherlands, access to disability insurance has become gradually more limited and demanding. The same holds for sheltered work. This puts disabled workers, and especially those with no working history, in disadvantaged competition with the non-disabled segment of the working population. In this study, we employ a Lotka–Volterra competition model based on differential equations to investigate how government expenditures for disability and other social benefits interacted in the period between 2010 and 2018. We contribute to the literature by showing that public expenditure for disability is not autonomous and that its competitive power in the social protection ecosystem changes both in type and over time. Our findings suggest that government expenditure for disability benefits in the Netherlands behaved both as prey in favour and predator at the cost of social exclusion and unemployment benefits. Reforms that took place in the ecosystem of interest are used to interpret such behaviour. [ABSTRACT FROM AUTHOR]

We consider fully discrete finite element approximations for a semilinear optimal control system of partial differential equations in two cases: for distributed and Robin boundary control. The ecological predator-prey optimal control model is approximated by conforming finite element methods mimicking the spatial part, while a discontinuous Galerkin method is used for the time discretization. We investigate the sensitivity of the solution distance from the target function, in cases with smooth and rough initial data. We employ low, and higher-order polynomials in time and space whenever proper regularity is present. The approximation schemes considered are with and without control constraints, driving efficiently the system to desired states realized using non-linear gradient methods. [ABSTRACT FROM AUTHOR]

In this paper, we analyzed the local stability of three species in two fractional tritrophic systems, with Caputo's fractional derivative and Holling type II and III functional responses, when the prey density has a linear growth. To begin, we obtained the equilibria in the first octant under certain conditions for the parameters. Subsequently, through linearization and applying the Routh-Hurwitz Criterion, we concluded that only the system with Holling type III exhibits an asymptotically stable equilibrium point, where the fractional derivative order belongs to the interval (0; 1]. Finally, we obtained the solution of the system with the Holling type III functional response, using the multistage homotopic perturbation method, and presented an example that shows the dynamics of the solutions around the stable equilibrium point. [ABSTRACT FROM AUTHOR]

In this paper, we analyzed the local stability of three species in two fractional tritrophic systems, with Caputo's fractional derivative and Holling type Ⅱ and Ⅲ functional responses, when the prey density has a linear growth. To begin, we obtained the equilibria in the first octant under certain conditions for the parameters. Subsequently, through linearization and applying the Routh-Hurwitz Criterion, we concluded that only the system with Holling type Ⅲ exhibits an asymptotically stable equilibrium point, where the fractional derivative order belongs to the interval $ (0, 1] $. Finally, we obtained the solution of the system with the Holling type Ⅲ functional response, using the multistage homotopic perturbation method, and presented an example that shows the dynamics of the solutions around the stable equilibrium point.

The paper focuses on the Lotka-Volterra function in its discrete form. The purpose of the study was to determine the fixed points of the function. The study employs the Banach Fixed Point Theorem and Contraction Mapping in Metric Space on the function to demonstrate the uniqueness of the fixed points and its continuous stability after several iterations, using the fixed points as the initial conditions. The study has shown that (0, 0), (0, α−1/β), (1+γ /δ, 0 ) and (1+γ/ δ, α−1/ β)are the fixed points of the function, with the initial pair serving as a trivial one and the other three solely depending on the parameter values for the behavior of the function. The outcome of the limit points of the function as the fixed points after several iterations forms a fixed orbit structure of the function, irrespective of the value of the parameter. The study also showed the uniqueness of the fixed points, demonstrating the stability and continuity of the function in its steady state. [ABSTRACT FROM AUTHOR]

The global trade network can be viewed from different perspectives. The common ones are the volume of exports and the number of countries that export products. The variation in number of exporting countries for products has been studied over the years. The U.S. dollar has been used for invoicing exportation in global trade, and its variation in values against global currencies negatively affects the volume of exports and the number of exporters. United States dollar indices (USDX) have been globally used to measure the values of global currencies against the U.S. dollar. Many studies have been done on the effect of the global economic crisis on global trade networks. However, scholars have not yet developed a complex dynamical model and established the critical transitional or threshold values that separate the resilient network of exporting countries from a non-resilient one. This study modelled the dynamic representation of a complex network of exporting countries by modifying Lotka-Volterra models. Using the developed model and empirical data USDX and export data of 161 countries from 1971 to 2020, the critical transition values have been calculated for different phases of economic crises. This study references five major crises, revealing that the variation of U.S. dollar indices significantly impacts global network system resilience. It suggests that policies reducing trade restrictions among trading partners are more important than controlling variation in U.S. dollar indices. This information will be crucial in addressing future crises like the Russian-Ukraine conflict.

During the last decades, macroecology has identified broad-scale patterns of abundances and diversity of microbial communities and put forward some potential explanations for them. However, these advances are not paralleled by a full understanding of the dynamical processes behind them. In particular, abundance fluctuations of different species are found to be correlated, both across time and across communities in metagenomic samples. Reproducing such correlations through appropriate population models remains an open challenge. The present paper tackles this problem and points to sparse species interactions as a necessary mechanism to account for them. Specifically, we discuss several possibilities to include interactions in population models and recognize Lotka-Volterra constants as a successful ansatz. For this, we design a Bayesian inference algorithm to extract sets of interaction constants able to reproduce empirical probability distributions of pairwise correlations for diverse biomes. Importantly, the inferred models still reproduce well-known single-species macroecological patterns concerning abundance fluctuations across both species and communities. Endorsed by the agreement with the empirically observed phenomenology, our analyses provide insights into the properties of the networks of microbial interactions, revealing that sparsity is a crucial feature. [ABSTRACT FROM AUTHOR]

The sign of the traveling wave speed in the two species Lotka-Volterra competition-diffusion system under strong competition provides important information about the competitive advantage of the species. To study the sign problem, we propose a mini-max characterization of the coefficients in the reaction terms when the wave speed is zero. Applying our characterization, some explicit conditions under which the speed sign can be determined are obtained. [ABSTRACT FROM AUTHOR]

The Lotka–Volterra predator–prey system for dynamic sea turtle population is solved using r-PINN-Adam method, a novel approach which combines Physics-Informed Neural Network (PINN) with restarting strategy. This method allows us to monitor the loss function values of PINN such that when there is no progress made, we stop the process and pick a good value to be used in the next process. Subsequently, the training time decreases and the accuracy increases. The numerical solutions are compared to the popular Runge–Kutta method in terms of correctness which presented graphically. Simulation results also displayed in terms of trainable parameters and optimal loss function performance. The research highlights the robustness and superiority of the proposed method, positioning it as a valuable tool for sea turtle conservation efforts.

New classes of conditionally integrable systems of nonlinear reaction–diffusion equations are introduced. They are obtained by extending a well-known nonclassical symmetry of a scalar partial differential equation to a vector equation. New exact solutions of nonlinear predator–prey systems with cross-diffusion are constructed. Infinite dimensional classes of exact solutions are made available for such nonlinear systems. Some of these solutions decay towards extinction and some oscillate or spiral around an interior fixed point. It is shown that the conditionally integrable systems are closely related to the standard diffusive Lotka–Volterra system, but they have additional features. [ABSTRACT FROM AUTHOR]

Every living organism is a physical, complex system which can be modeled by nonlinear dynamical equations in some very narrowed cases. Here we discuss the adoption and potential application of Lotka–Volterra equations (with damping) to simulate, on a very general level, an organism's response to a dose of ionizing radiation. The step-by-step calculations show how such modeling can be applied to practically every living thing affected by some external stressor. It is presented that Lotka–Volterra prey–predator equations can successfully model the homeostasis (equilibrium) state of the living matter, with balance between detrimental and beneficial factors which interact in the system. It was shown that too large of a radiation dose can break the damping process, making the system unstable, which is analogous to the irreversible transformation of the irradiated cell/organism. On the contrary, too low of a radiation dose makes the damping factor slightly negative, which means that some nonzero low level of ionizing radiation is the most optimal for an organism's homeostasis. [ABSTRACT FROM AUTHOR]

We characterize when the unique equilibrium point of the positive quadrant of the twodimensional Lotka–Volterra system is a global attractor in that quadrant. Additionally, we classify the phase portraits of this class of Lokta–Volterra system. [ABSTRACT FROM AUTHOR]

Moose present a complex management problem because they generate a mixture of benefits and costs to humans, some of which are caused by browsing of regenerating trees. We developed a Lotka–Volterra model, parameterized by moose management areas, to link moose browsing to spruce and balsam fir dynamics on the island of Newfoundland. The model predicts the distribution of moose, adult fir, and spruce well. Empirical estimates of juvenile fir biomass were variable, and our model predicted its biomass poorly. Our model predicts a small negative effect of moose on adult fir biomass (−0.06%) and juvenile fir biomass (−1.65%) and a small positive effect on spruce biomass (+0.02%) under baseline assumptions, but larger effects (±10%–60%) if moose browse commercial softwoods preferentially. Small effects of moose on trees at steady state do not fully reflect the importance of moose because moose parameters (e.g., growth rate and harvesting rate) impacted the return time of our model from disturbance. Our model analysis demonstrates one way to add animal effects into vegetation growth models and suggests that parameterizing ecological models by management unit is useful when the data to support more detailed models are not available. [ABSTRACT FROM AUTHOR]

This paper is concerned with the two-species chemotaxis-competition model with degenerate diffusion, { u t = Δ u m 1 − χ 1 ∇ ⋅ (u ∇ w) + μ 1 u (1 − u − a 1 v) , x ∈ Ω , t > 0 , v t = Δ v m 2 − χ 2 ∇ ⋅ (v ∇ w) + μ 2 v (1 − a 2 u − v) , x ∈ Ω , t > 0 , 0 = Δ w + u + v − M ‾ (t) , x ∈ Ω , t > 0 , with ∫ Ω w (x , t) d x = 0 , t > 0 , where Ω : = B R (0) ⊂ R n (n ≥ 5) is a ball with some R > 0 ; m 1 , m 2 > 1 , χ 1 , χ 2 , μ 1 , μ 2 , a 1 , a 2 > 0 ; M ‾ (t) is the spatial average of u + v . In this paper, we show that if m 1 < 2 − 4 n , χ 1 > n (2 − m 1) n (2 − m 1) − 4 ⋅ max { 1 , a 1 } μ 1 and χ 2 > μ 2 a 2 or m 2 < 2 − 4 n , χ 1 > μ 1 a 1 and χ 2 > n (2 − m 2) n (2 − m 2) − 4 ⋅ max { 1 , a 2 } μ 2 , then there exist radially symmetric initial data such that the weak solution blows up in finite time in the sense that there is T ˜ max ∈ (0 , ∞) such that lim sup t ↗ T ˜ max (∥ u (t) ∥ L ∞ (Ω) + ∥ v (t) ∥ L ∞ (Ω)) = ∞. To obtain this result, we apply the method in the previous paper (Discrete Contin. Dyn. Syst., Ser. B 28(1):262–286, 2023) to derive an integral inequality for a moment-type functional, which was introduced by Winkler (Z. Angew. Math. Phys. 69(2):69, 2018). Moreover, before proving blow-up of solutions in the above model, we give a result on finite-time blow-up under the same conditions for m 1 , m 2 , χ 1 and χ 2 in the model with the terms Δ u m 1 , Δ v m 2 replaced with the nondegenerate diffusion terms Δ (u + ε) m 1 , Δ (v + ε) m 2 , where ε ∈ (0 , 1 ] . [ABSTRACT FROM AUTHOR]

This research employs a modified version of the Lotka-Volterra non-linear first-order ordinary differential equations to model and analyze the parasitic impact of ticks on dogs. The analysis reveals that fluctuations in pesticide effects significantly influence tick populations and the size of the canine host. The study also uncovers that alterations in the size of the interacting species can lead to both stable and unstable states. Interestingly, in a pesticide-free environment, a decline in the inter-competition coefficient catalyzes an increase in the sizes of both interacting species. This increase, although marginal for the tick population, contributes to overall system stability. The findings underscore the utility of the Lotka-Volterra non-linear first-order ordinary differential equations in modeling the parasitic effect of ticks on dogs. To protect pets, particularly dogs, from the harmful effects of tick infestation, this study recommends the appropriate and regular application of disinfectants. [ABSTRACT FROM AUTHOR]

A sustainable and cost-effective approach of waste water management is biological treatment for reducing organic carbon, nitrate, and phosphate content. Co-culturing of algae with bacteria in wastewater leads to higher biomass yield and improvement in COD/nutrients removal compared to the single strain counterparts. In this study, a mathematical modeling framework is proposed to predict the dynamic behavior of microbial co-culture in dairy waste water. Initially, the model has been developed to predict the biomass growth and COD/nutrients removal with discrete cultures (algae and bacteria). As an extension of the single strain kinetic model, Lotka–Volterra model was formulated to explore the symbiotic relationship between algae and bacteria in a co-culture and the impact of the interactions on the COD/nutrients removal efficiency and growth dynamics. Supporting experiments were carried out in 6 parallel sets (3 sets with triplicates) with standalone algae (Chlorella vulgaris, CV), bacteria (activated sludge), and co-culture in real-time dairy liquid effluent in lab flasks and predicted values from modeling were validated against experimental findings. Statistical analysis confirms reasonably good agreement between the model predictions and experimental findings indicating a positive synergistic effect of the algae–bacterial co-culture on COD removal. [ABSTRACT FROM AUTHOR]

This article is devoted to the analysis of a fractional chemotaxis model in RN with a time fractional variation in the Caputo sense and a fractional spatial diffusion. This model encompasses the fractional Keller-Segel system [9] which describes the movement of living organisms towards higher concentration regions of chemical attractants, and a fractional Lotka-Volterra competition model [16] describing the competition interspecies in which one of the competing species avoids encounters with rivals by means of chemorepulsion. We prove product estimates in Besov-Morrey spaces and derive global estimates for mild solutions of the fractional heat equation. We use these results to prove the existence and uniqueness of global mild solutions for the differential system in a framework of Besov-Morrey spaces. [ABSTRACT FROM AUTHOR]

The objective of this work is to model a population management in which two species intervene. There is an interdependence predator-prey relationship between them. It will be assumed, as a descriptive biological model, the Lotka-Volterra one with the application of an Allee effect in one of the dynamics of the species. This incorporation into the system make the model more realistic. In view of the advantages, the model will be in its fractional version, that is, where derivatives of Caputo of fractional order are considered, with the fractional order in (0,1]. Existence and uniqueness of the model solution will be explicitly proved, and a non-negative invariance of the solution and the stability of the resulting equilibrium will be studied. In order to solve the problem, the use of fractional numerical techniques will be of fundamental importance and absolutely essential to conclude the analysis. Some examples of great importance will be shown. [ABSTRACT FROM AUTHOR]

The airways of people with cystic fibrosis (CF) often harbor a diverse microbiota and in recent years, much effort has been invested in cataloguing these. In spite of providing a wealth of insight, this cataloguing tells us little about how the organisms interact with one another in the CF airways. However, such relationships can be inferred using the theoretical framework of the Lotka-Volterra (LV) model. In the current work, we use a generalized Lotka-Volterra model to interrogate the nationwide data collected and curated by the UK CF Registry. This longitudinal dataset (covering the period 2008-2020) contains annual depositions that record the presence/absence of microbial taxa in each patient, their medication, and their CF genotype. Specifically, we wanted to identify trends in ecological relationships between the CF microbiota at a nationwide level, and whether these are potentially affected by medication. Our results show that some medications have a distinct influence on the microbial interactome, especially those that potentially influence the "gut-lung axis" or mucus viscosity. In particular, we found that patients treated with a combination of antimicrobial agents (targeting the airway microbiota), digestive enzymes (assisting in the assimilation of dietary fats and carbohydrates), and DNase (to reduce mucus viscosity) displayed a distinctly different airway interactome compared with patients treated separately with these medications. [ABSTRACT FROM AUTHOR]

Population dynamics of migratory species can be modeled using spatial bipartite networks in which nodes representing breeding and winter regions are distributed longitudinally and connected by links representing migratory movements. Understanding the factors that influence the connectivity and population size of such networks is important for effective conservation. In migration ecology terminology, strong migratory connectivity is a network with low node degree and weak or diffuse connectivity is a network with high node degree. We present a model of migration networks using a Lotka-Volterra system of differential equations in which each winter-breeding link is represented as a subpopulation which competes with other subpopulations that share breeding or winter regions. We analyze how habitat distribution and relative costs of migration paths affect the coexistence of subpopulations and hence, the connectivity pattern (weak, moderate or strong). We find that, in the absence of dispersal among subpopulations, strong connectivity occurs when winter habitat has the same longitudinal distribution as breeding habitat, and/or costs of cross-longitudinal migration are high. Moderate connectivity arises otherwise. Total population size is maximized when each longitudinal region has the same amount of breeding habitat as winter habitat and decreases with the costs of migration. Including dispersal leads to weak connectivity and reduces population size. Our results suggest that actions that conserve habitat so as to preserve matching habitat distributions and minimize costs of migration would be most effective. [ABSTRACT FROM AUTHOR]

A two-dimensional cubic Lotka-Volterra system depending on two parameters is considered. Local dynamics in a neighbourhood of the origin of the phase plane, when the parameters lay in a sufficiently small neighbourhood of the origin, is investigated. The study is performed when some additional hypotheses on the coefficients are satisfied. From one up to four different equilibria and several types of codimension one local bifurcations are found. For each of the identified cases, bifurcation diagrams are given. [ABSTRACT FROM AUTHOR]

In this paper, we consider the dynamical behaviors of a Lotka–Volterra competition‐diffusion‐advection system with Dirichlet boundary conditions in a one‐dimensional homogeneous environment, where the two competing species have the same growth and advection rates but different random dispersal rates. Firstly, We determine the existence and uniqueness of positive steady states for the single species model by investigating the properties of the corresponding principal eigenvalues. Secondly, some conditions are provided to assert the globally asymptotical stability of semi‐steady states for the two species competitive model. The results suggest that: (a) the species will prevail whenever it exists if the diffusion rate of the other species is small or large enough; (b) the species with much faster diffusion rate will win in the long run if the diffusion rates of the two species fall into some particular ranges. Finally, we show that if the appropriate diffusion rates are selected, then the two species will coexist. Our results indicate that both "competition exclusion principle" and "co‐existence" may happen depending on the different dispersal strategies. [ABSTRACT FROM AUTHOR]

We use a system dynamics model to analyse the cycle of production of a nonrenewable natural resource, with a specific interest in crude oil. This subject has been empirically studied for a long time. However, modelling studies able to correlate the peaking with the parameters of the system have been very rare, and only recently proposed. In the present paper, we examine the timing of the peaking mainly as a function of the energy return for energy invested (EROI). The model provides approximate formulas for evaluating the peak time and "rules of thumb" that are useful for understanding the peaking phenomenon in the exploitation of natural resources. It shows that the peaking of the production curve occurs at a time that is inversely proportional to the EROI of the process at the start of the cycle. [ABSTRACT FROM AUTHOR]

The global increase in consumption of plastics has caused serious environmental problems. When discarded in nature, these materials degrade into microplastic particles, which have spread around the world and caused serious harm to different ecosystems. For instance, their presence in living organisms can alter predator-prey dynamics. Huang et al. [1] successfully applied a modified Lotka-Volterra model to study the impact of microplastic particles on population dynamics of predators and preys. Here, we propose to simplify their model, reducing its number of parameters without compromising its descriptive and predictive power. Also, we used time-varying intraspecific coefficients and solved analytically particular situations characterized by extreme effects on predatory performance, in addition to assigning a different meaning to the parameters removed from the original model. In addition, we have presented two important characteristic times related to the faster accumulation of microplastic particles in predators than in prey. This simpler description should be used to avoid misleading comparison between systems with microplastic particles and the ones free of them. [ABSTRACT FROM AUTHOR]

Population and community ecology traditionally has a very strong theoretical foundation with well‐known dynamical models, such as the logistic and its variations, and many modifications of the classical Lotka–Volterra predator–prey and interspecific competition models. More and more, these classical models are being confronted with data via fitting to empirical time series for purposes of projections or for estimating model parameters of interest. However, using statistical models to fit theoretical models to data is far from trivial, especially for time series data where subsequent measurements are not independent. This raises the question of whether statistical inferences using pure observation error models, such as simple (non‐)linear regressions, are biased, and whether more elaborate process error models or state‐space models have to be used to address this complexity. In order to help empiricists, especially researchers working with experimental laboratory populations in micro‐ and mesocosms, make informed decisions about the statistical formalism to use, we here compare different error structures one could use when fitting classical deterministic ordinary differential equation (ODE) models to empirical data. We consider a large range of biological scenarios and theoretical models, from single species to community dynamics and trophic interactions. In order to compare the performance of different error structure models, we use both realistically simulated data and empirical data from microcosms in a Bayesian framework. We find that many model parameters can be estimated precisely with an appropriate choice of error structure using pure observation error or state‐space models, if observation errors are not too high. However, Allee effect models are typically hard to identify and state‐space models should be preferred when model complexity increases. Our work shows that, at least in the context of low environmental stochasticity and high quality observations, deterministic models can be used to describe stochastic population dynamics that include process variability and observation error. We discuss when more complex state‐space model formulations may be required for obtaining accurate parameter estimates. Finally, we provide a comprehensive tutorial for fitting these models in R. [ABSTRACT FROM AUTHOR]

Abstract Population and community ecology traditionally has a very strong theoretical foundation with well‐known dynamical models, such as the logistic and its variations, and many modifications of the classical Lotka–Volterra predator–prey and interspecific competition models. More and more, these classical models are being confronted with data via fitting to empirical time series for purposes of projections or for estimating model parameters of interest. However, using statistical models to fit theoretical models to data is far from trivial, especially for time series data where subsequent measurements are not independent. This raises the question of whether statistical inferences using pure observation error models, such as simple (non‐)linear regressions, are biased, and whether more elaborate process error models or state‐space models have to be used to address this complexity. In order to help empiricists, especially researchers working with experimental laboratory populations in micro‐ and mesocosms, make informed decisions about the statistical formalism to use, we here compare different error structures one could use when fitting classical deterministic ordinary differential equation (ODE) models to empirical data. We consider a large range of biological scenarios and theoretical models, from single species to community dynamics and trophic interactions. In order to compare the performance of different error structure models, we use both realistically simulated data and empirical data from microcosms in a Bayesian framework. We find that many model parameters can be estimated precisely with an appropriate choice of error structure using pure observation error or state‐space models, if observation errors are not too high. However, Allee effect models are typically hard to identify and state‐space models should be preferred when model complexity increases. Our work shows that, at least in the context of low environmental stochasticity and high quality observations, deterministic models can be used to describe stochastic population dynamics that include process variability and observation error. We discuss when more complex state‐space model formulations may be required for obtaining accurate parameter estimates. Finally, we provide a comprehensive tutorial for fitting these models in R.

Diagonal linear networks (DLNs) are a toy simplification of artificial neural networks; they consist in a quadratic reparametrization of linear regression inducing a sparse implicit regularization. In this paper, we describe the trajectory of the gradient ow of DLNs in the limit of small initialization. We show that incremental learning is effectively performed in the limit: coordinates are successively activated, while the iterate is the minimizer of the loss constrained to have support on the active coordinates only. This shows that the sparse implicit regularization of DLNs decreases with time. This work is restricted to the underparametrized regime with anti-correlated features for technical reasons. [ABSTRACT FROM AUTHOR]

Predator-prey models are often used in ecology to represent realistic encounters between species. Lotka-Volterra differential equations are famously used by ecologists to model the relationship between a predator and prey. Given ecological data collection is known to be time consuming and difficult, the purpose of this paper is to optimize the process and improve sampling efficiency. The method of sequential optimality is applied to a more complex population model with multiple basins of attraction, the Lotka-Volterra differential equations. In this application, various theoretical scenarios are simulated using varying sample sizes and timeframes to determine optimal designs. The first scenario uses a standard sample size and design window to represent a realistic budget for ecological data collection. The second scenario decreases the budget and extends the design window as a representation of a limited budget. The third scenario represents ample resources available for sampling. The three scenarios are used to then compare the optimal designs and help ecologists evaluate the method of sequential optimality. I, A and D optimal designs are compared in the simulation results. The I-optimality criterion tends to outperform the A and D criteria and is recommended as a primary selection when using the sequential optimality algorithm. This simulation study has further developed sequential optimality by predicting optimal sampling regimes according to the dynamics associated with the Lotka-Volterra differential equations. [ABSTRACT FROM AUTHOR]

This paper deals with the two-species chemotaxis-competition models u t = d 1 Δ u - χ 1 ∇ · (u ∇ w) + μ 1 u (1 - u κ 1 - 1 - a 1 v λ 1 - 1) , x ∈ Ω , t > 0 , v t = d 2 Δ v - χ 2 ∇ · (v ∇ w) + μ 2 v (1 - a 2 u λ 2 - 1 - v κ 2 - 1) , x ∈ Ω , t > 0 , 0 = d 3 Δ w + α u + β v - h (u , v , w) , x ∈ Ω , t > 0 , where Ω ⊂ R n (n ≥ 2) is a bounded domain with smooth boundary, and h = γ w or h = 1 | Ω | ∫ Ω (α u + β v) d x . In the case that κ 1 = λ 1 = κ 2 = λ 2 = 2 and h = γ w , it is known that smallness conditions for the chemotactic effects lead to boundedness of solutions in, for example, [21]. However, the case that the chemotactic effects are large seems not to have been studied yet; therefore, it remains to consider the question whether the solution is bounded also in the case that the chemotactic effects are large. The purpose of this paper is to give a negative answer to this question. [ABSTRACT FROM AUTHOR]

The paper is concerned with a singular limit for the bistable traveling wave problem in a very large class of two-species fully nonlinear parabolic systems with competitive reaction terms. Assuming existence of traveling waves and enough compactness, we derive and characterize the limiting problem. The assumptions and results are discussed in detail. The free boundary problem obtained at the limit is specified for important applications.

Every living organism is a physical, complex system which can be modeled by nonlinear dynamical equations in some very narrowed cases. Here we discuss the adoption and potential application of Lotka–Volterra equations (with damping) to simulate, on a very general level, an organism’s response to a dose of ionizing radiation. The step-by-step calculations show how such modeling can be applied to practically every living thing affected by some external stressor. It is presented that Lotka–Volterra prey–predator equations can successfully model the homeostasis (equilibrium) state of the living matter, with balance between detrimental and beneficial factors which interact in the system. It was shown that too large of a radiation dose can break the damping process, making the system unstable, which is analogous to the irreversible transformation of the irradiated cell/organism. On the contrary, too low of a radiation dose makes the damping factor slightly negative, which means that some nonzero low level of ionizing radiation is the most optimal for an organism’s homeostasis.

Population dynamics of a predator-prey system is usually simulated by the classical Lotka-Volterra models, which were successfully applied to the population dynamics of snowshoe hare and lynx and many other predator-prey systems. Attempts were made to apply them also to insect predator-prey systems, but in terms of biological control, they did not reveal the features of the predators that control the abundance of their prey. The most conspicuous example of failure of Lotka-Volterra models applied to insect predator-prey systems are ladybird-aphid systems, in which these models usually fail to fit empirical data. Because of their practical importance and because they are very well studied, we have chosen aphid-ladybird systems as a model. We summarize the results published on various aspects of the population dynamics of aphid-ladybird systems and present them in the context of empirical data. Using new data, we more closely specify the existing metapopulation model of aphid-ladybird interactions. Based on the arguments presented here, we conclude that the ladybird-aphid case can be generalized to insect (and maybe even other) predator-prey systems, where the ratio of the generation times of the predator to that of the prey (GTR) is large. In such systems, the main selection pressure on predators is choosing the best strategy to maximize survival of their offspring, rather than on maximization of the amount of prey eaten. Thus voracity, which is the main determinant of population dynamics in Lotka-Volterra models, loses its role and is replaced by optimization of the choice of oviposition sites in systems with large GTRs.

This paper provides an approximate analytic solution to the three species Lotka – Volterra differential equations by symbolic regression. The approximate analytic solution through symbolic regression is made as close as desired to the actual analytic solution by using the Jacobian system. This is proposed as the equilibrium will be stabilized if and only if the real parts of each of the eigenvalues are negative. As a result, the symbolic regression approach is found to provide an approximation to the faster convergence that can be expected with a more refined Euler numerical approach.