Lotka-Volterra Research Articles

Global bifurcation analysis and derivation of an optimal harvesting policy of a prey–predator model with Holling type-IV functional scheme

ABSTRACT This paper establishes a prey–predator model with a Holling type IV functional response and proportional harvesting in prey species and nonlinear harvesting in predator species. We provide a completely local and global bifurcation analysis of the model to characterize some meaningful results that may emerge from the interaction of biological resources. The proposed model exhibits various complex behaviors involving heteroclinic and homoclinic orbits, and we prove analytically that it also undergoes transcritical, Hopf, and Bogdanov–Takens bifurcations in the prey–predator plane. Moreover, the system is unfolded to compute the bifurcation curves, highlighting the critical dynamics by perturbing two bifurcation parameters near the cusp. Furthermore, from the economic point of view, the bionomic equilibrium of the system is studied, and optimal harvesting policy is derived using the Pontryagin’s maximum principle. Numerical simulations are presented to verify our analytical results.

Multiple Time Scales and Normal Form of Hopf Bifurcation in a Delayed Reaction–Diffusion–Advection System

In the reaction–diffusion systems, the Laplacian is usually a self-adjoint operator. Incorporating advection terms generally destroys this and yields a quite complicated decomposition of phase space in the Center Manifold Reduction (CMR) method. Considering that the Multiple Time Scales (MTS) method can be directly applied to bifurcation analysis and normal form derivation based on algebraic operations, and the computational process is relatively universal, we apply the MTS method to analyze the Hopf bifurcation problem of reaction–diffusion–advection systems with time delay. First, we introduce the MTS method for the system with a specific boundary condition, which is used to derive the normal form of Hopf bifurcation. Calculating the normal form using the MTS method can be summarized as determining the bifurcation parameters, conducting Taylor expansion based on the MTS idea, and eliminating secular terms. Then, the key coefficients in the normal form are explicitly given, which are also compared with the results obtained by the CMR method, and both methods lead to the same bifurcation results. Finally, we use the MTS method to calculate the normal form of Hopf bifurcation for a predator–prey model with mixed boundary conditions. We find that the spatially nonhomogeneous periodic solutions appear near the equilibrium in the system when the time delay exceeds the critical value, and the theoretical results are illustrated by numerical simulations.

Who is afraid of modelling time as a continuous variable?

Abstract Most models of ecological and eco‐evolutionary processes involve creating trajectories of something, be it population densities, average trait values, or environmental states, over time. This requires decision‐making regarding how to represent the flow of time in models. Most ecologists are exposed to continuous‐time models (typically in the form of ordinary differential equations) as part of their training, especially since the famous Lotka‐Volterra predator–prey dynamics are formulated this way. However, few appear sufficiently well trained to produce their own work with continuous‐time models and may lack exposure to the true versatility of available methods. Specifically, knowledge that discrete individuals can be modelled in continuous time using the Gillespie algorithm is not as widespread as it should be. I will illustrate the flexibility of continous‐time modelling methods such that researchers can make informed choices, and not resort to discretizing time as a ‘default’ without a clear biological motivation to do so. I provide three example‐based tutorials: (1) a comparison of deterministic and stochastic dynamics of the Lotka‐Volterra predator–prey model, (2) an evaluation of matelessness in a hypothetical insect population (and of selection to mate more often by either searching more efficiently or by shortening the ‘time out’ after each mating) and (3) within‐season density dependence followed by a birth pulse leading to Beverton‐Holt or Ricker dynamics depending on whether the deaths of conspecifics help reduce the mortality of others or not (compensatory mortality). I highlight properties of the exponential distribution that, while counter‐intuitive, are good to know when deriving expected lifetime reproductive success or other similar quantities. I also give guidance on how to proceed if the so‐called memorylessness assumption does not hold in a given situation, and show how continuous and discrete times can be freely mixed if the biological situation dictates this to be the preferred option. Continuous‐time models can also be empirically fitted to data, and I review briefly the insight this gives into the so‐called ‘do hares eat lynx?’ paradox that has been plaguing the interpretation of the Hudson Bay hare and lynx dataset.

Dynamic Complexity of a Discrete-time Prey-Predator Model with Mixed Functional Responses

Plastic particles less than 5 mm in size, known as microplastics, can infiltrate the environment from a variety of sources. This study aimed to assess the presence of microplastics (MPs) in the atmosphere at fifteen major traffic intersections in the greater Dhaka city from July to August 2022.The deposition rate (DP) of MPs was calculated and the observed DPs value ranged from 1.11×106 MPs/m2/day to 5.78×106 MPs/m2/day with a portion of microplastics exhibiting fluorescence activity. Fourier Transform Infrared (FTIR) spectroscopy was used to analyze the functional groups present in microplastics. The identified polymer compositions of MPs were polyethylene (PE), Nylon-6 and natural polymers (cellulose and rubber). The current study highlights the data and knowledge gaps on the atmospheric transport of microplastics and their impact to the deterioration of overall urban air quality. Dhaka Univ. J. Sci. 72(2): 15-24, 2024 (July)

Mesopredators have differing influences on prey habitat use and diel activity in a multipredator landscape

AbstractResource distribution, habitat structure, and predators greatly influence spatial and temporal landscape use by prey species. The “risky places” hypothesis establishes prey will proactively respond to predators' presence based on habitat cues, whereas the “risky times” hypothesis predicts prey will reactively respond by increasing vigilance in the presence of predators regardless of habitat cues. We fit a multiscale, Bayesian species interaction occupancy model with detection/non‐detection data to evaluate black‐tailed jackrabbit (Lepus californicus) and eastern cottontail rabbit (Sylvilagus floridanus) habitat use in the presence and absence of coyotes (Canis latrans), American badgers (Taxidea taxus), and swift foxes (Vulpes velox). We also evaluated how species‐specific predator presence modified temporal activity patterns of prey. Jackrabbits decreased habitat use in areas with greater forage and opted to use areas with greater visibility when coyotes or swift foxes were present. However, cottontails used habitat in open areas with greater visibility when American badgers were present and all other predators absent, suggesting dissimilar habitat‐use patterns dictated by predator‐specific risks. Both lagomorph species are nocturnal with segregated peaks of activity compared with predators, suggesting fine‐scale temporal use partitioning. Our results provide insights into predator–prey dynamics across heterogenous landscapes in a multi‐predator system.

Stability Analysis of a T-S Prey-Predator Model with Disease in both Species using Fuzzy Impulsive Control

Stability Analysis of a T-S Prey-Predator Model with Disease in both Species using Fuzzy Impulsive Control

Existence of positive periodic solutions for nonlinear Lotka-Volterra delay dynamic systems on time scales

Existence of positive periodic solutions for nonlinear Lotka-Volterra delay dynamic systems on time scales

Time-dependent probability density functions and information geometry in a stochastic prey–predator model of fusion plasmas

Time-dependent probability density functions and information geometry in a stochastic prey–predator model of fusion plasmas

Local and global dynamics of a prey–predator system with fear, Allee effect, and variable attack rate

Fear prompts prey to adopt risk-averse behaviors, such as reduced foraging activity, increased vigilance, and avoidance of areas with high predator presence, which affects its reproduction. In a real scenario, a population requires a minimum density to avoid extinction, known as an Allee threshold. In light of these biological factors, we propose a predator–prey model with (i) a fear effect in a prey population, (ii) an Allee effect in a predator population, and (iii) a non-constant attack rate that modifies the functional response. We ensured the non-negativity and boundedness of the solutions and examined the local and global stability status for each existing steady state solutions. We investigated some deep dynamical properties of the system by varying different parameters, such as cost of fear in prey and strength of the Allee effect in predators and their mortality rate. In codimension one bifurcations, we observed saddle node, Hopf, homoclinic, and coalescence of two limit cycles. Additionally, codimension two bifurcations were observed, including Bautin and Bogdanov Takens bifurcations. To provide a clearer understanding of these bifurcations, we conducted biparametric analysis involving the fear and Allee parameters, as well as the fear parameter and predator mortality rate. Our investigation shows that cost of fear and strength of Allee strongly influences the survival status of the predator. Furthermore, bistability and tristability reveal that the survival and extinction of predator are dependent on the initial population level. Numerical simulations and graphical illustrations are provided to support and validate our theoretical findings.

Nonlinear dynamics in a fear‐driven predator–prey system: Bistability, bifurcations, hydra effect, and optimal harvesting

The impact of predator‐driven fear on ecosystems is significant and can encompass both trophic (direct) and nontrophic (indirect) effects. Previous studies have shown that nontrophic fear effects have an important role in predator–prey dynamics. This study investigates the nontrophic fear effect on prey caused by generalist predators and explores optimal harvesting. We assume that the reproduction rate of prey is reduced due to the fear effect, and generalist predator follows Holling type II foraging strategy for predation. Additionally, we assume that predators are commercially valuable and harvested proportionately to their density. We demonstrate the existence of equilibrium points, their local and global stability, and bifurcation analysis. We observe that the model system undergoes a sequence of codimension one and codimension two bifurcations. Our results show that in the absence of predator harvesting, increasing levels of fear destabilize the predator–prey system, and controlled harvesting is beneficial for the coexistence of both populations. Also, the harvesting of predators may produce hydra and multiple hydra effects. We identify different types of bistability phenomena, which emphasize the importance of initial population size. Further, an optimal harvesting policy is also explored by formulating an optimal control problem (OCP). The harvesting cost‐functional is designed by incorporating the bionomic equilibrium state. We use Pontryagin's maximum principle and solve the OCP numerically. It is observed that implementing optimal harvesting not only contributes to the ecological benefits by maintaining a sustainable balance of predator–prey evolution but also plays a significant role in maximizing the economic benefits.

Bifurcation structure of steady states for a cooperative model with population flux by attractive transition

AbstractThis paper studies the steady states to a diffusive Lotka–Volterra cooperative model with population flux by attractive transition. The first result gives many bifurcation points on the branch of the positive constant solution under the weak cooperative condition. The second result shows every steady state approaches a solution of the scalar field equation as the coefficients of the flux tend to infinity. Indeed, the numerical simulation using pde2path exhibits the global bifurcation branch of the cooperative model with large population flux is near that of the scalar field equation.

Learning integral operators via neural integral equations

Nonlinear operators with long-distance spatiotemporal dependencies are fundamental in modelling complex systems across sciences; yet, learning these non-local operators remains challenging in machine learning. Integral equations, which model such non-local systems, have wide-ranging applications in physics, chemistry, biology and engineering. We introduce the neural integral equation, a method for learning unknown integral operators from data using an integral equation solver. To improve scalability and model capacity, we also present the attentional neural integral equation, which replaces the integral with self-attention. Both models are grounded in the theory of second-kind integral equations, where the indeterminate appears both inside and outside the integral operator. We provide a theoretical analysis showing how self-attention can approximate integral operators under mild regularity assumptions, further deepening previously reported connections between transformers and integration, as well as deriving corresponding approximation results for integral operators. Through numerical benchmarks on synthetic and real-world data, including Lotka–Volterra, Navier–Stokes and Burgers’ equations, as well as brain dynamics and integral equations, we showcase the models’ capabilities and their ability to derive interpretable dynamics embeddings. Our experiments demonstrate that attentional neural integral equations outperform existing methods, especially for longer time intervals and higher-dimensional problems. Our work addresses a critical gap in machine learning for non-local operators and offers a powerful tool for studying unknown complex systems with long-range dependencies.

Contrasting Dynamics of Intracellular and Extracellular Antibiotic Resistance Genes in Response to Nutrient Variations in Aquatic Environments.

The propagation of antibiotic resistance in environments, particularly aquatic environments that serve as primary pathways for antibiotic resistance genes (ARGs), poses significant health risks. The impact of nutrients, as key determinants of bacterial growth and metabolism, on the propagation of ARGs, particularly extracellular ARGs (eARGs), remains poorly understood. In this study, we collected microorganisms from the Yangtze River and established a series of microcosms to investigate how variations in nutrient levels and delivery frequency affect the relative abundance of intracellular ARGs (iARGs) and eARGs in bacterial communities. Our results show that the relative abundance of 7 out of 11 representative eARGs in water exceeds that of iARGs, while 8 iARGs dominate in biofilms. Notably, iARGs and eARGs consistently exhibited opposite responses to nutrient variation. When nutrient levels increased, iARGs in the water also increased, with the polluted group (COD = 333.3 mg/L, COD:N:P = 100:3:0.6, m/m) and the eutrophic group (COD = 100 mg/L, COD:N:P = 100:25:5, m/m) showing 1.2 and 3.2 times higher levels than the normal group (COD = 100 mg/L, COD:N:P = 100:10:2, m/m), respectively. In contrast, eARGs decreased by 6.7% and 8.4% in these groups. On the other hand, in biofilms, higher nutrient levels led to an increase in eARGs by 1.5 and 1.7 times, while iARGs decreased by 17.5% and 50.1% in the polluted and eutrophic groups compared to the normal group. Moreover, while increasing the frequency of nutrient delivery (from 1 time/10 d to 20 times/10 d) generally did not favor iARGs in either water or biofilm, it selectively enhanced eARGs in both. To further understand these dynamics, we developed an ARGs-nutrient model by integrating the Lotka-Volterra and Monod equations. The results highlight the complex interplay of bacterial growth, nutrient availability, and mechanisms such as horizontal gene transfer and secretion influencing ARGs' propagation, driving the opposite trend between these two forms of ARGs. This contrasting response between iARGs and eARGs contributes to a dynamic balance that stabilizes bacterial resistance levels amid nutrient fluctuations. This study offers helpful implications regarding the persistence of bacterial resistance in the environment.

Evaluating the tipping point of a complex system: The case of disruptive technology

AbstractComplex systems often operate in equilibriums that can be disrupted under specific conditions, driving the system to undergo an irreversible phase transition (aka tipping point) into a new equilibrium. Disruptive technology, as introduced in the book Innovator's Dilemma, is a new emerging technology that can successfully displace incumbent technologies and push the market through phase transition into a new equilibrium. In this paper, the authors model the market disruption caused by a disruptive technology as a complex system, with dynamics that show a phase transition or tipping point, after which the system shifts into a new equilibrium aiming at assessing the success or failure of a disruptive technology. The purpose of this study is to mathematically model a tipping point measure of complex networks and a theoretical framework for disruptive technology dynamics. A predator–prey model is used to emulate the behavior of a disruptive technology versus an incumbent technology, and a resilience index is integrated to measure the tipping point in the market where the disruptive technology will overtake the incumbent technology. This methodology is applied to a historical case study of film versus digital versus cell phone cameras, demonstrating the application of this methodology. The study's contribution can be applied to both systems engineering and disruptive innovation management. The proposed approach can help stakeholders assess the impact of a potential new disruptive technology and use the resulting resilience index as a measurement for adjusting technology requirements and systems management approaches to achieve a desired outcome.

Equlibrium of communities in the Lotka-Volterra model

Equlibrium of communities in the Lotka-Volterra model

Stationary distribution and mean extinction time in a generalist prey–predator model driven by Lévy noises

In this study, a delayed generalist prey–predator model incorporating the logistic-like growth and Holling type-III functional response for predator subjected to two Lévy noises is established. Such environmental fluctuations can describe abrupt change in population density. First, we conduct stability analysis and determine the critical value of time delay when Hopf bifurcation takes place. Then, the existence of the global positive solution and stability in distribution of stochastic model are studied. Numerical results are given to illustrate the analytical findings, and mainly dedicated to the consequences of time delay and Lévy noises on the stationary distribution, mean extinction time and reproduction rate of prey–predator populations, via stationary probability distribution function (PDF), mean first passage time (MFPT) and relaxation time, respectively. Our findings indicate that the decreases in time delay, noise intensities (Di,i=1,2), and stability indexes within the range of αi∈[1,2] are more likely to maintain the population in a high density state. Phase transition occurs when the system is in the suitable parameter regime. Additionally, there exist optimal noise intensities that slow down switching of prey species from a stable state to an extinction state. Smaller time delay τ and stability indexes αi inhibit population extinction, whereas smaller skewness parameter β speed up prey extinction. More importantly, two noises can tune the enhanced stability characteristic of the system.

Effects of diffusion and advection on predator-prey dynamics in an advective patchy environment

In this paper, we investigate a specialist predator-prey model within a closed patchy network of streams. Specifically, we focus on the dynamics and asymptotic profiles of positive steady states. Our findings reveal that specialist predators can successfully invade when the mortality rate remains sufficiently low. Additionally, we explore the effects of diffusion and advection on these steady states and the overall species concentration.

The present state of leech fauna (Annelida: Hirudinea) in Dal Lake, Jammu & Kashmir, India

Dal Lake supports a diverse group of fauna, including fishes, aquatic birds, zooplankton, and macrobenthic invertebrates. This study is the first taxonomic documentation of the leech fauna of Dal Lake. A total of nine leech taxa belonging to four families were identified and described: Glossiphoniidae – Alboglossiphonia weberi, A. heteroclita, Glossiphonia complanata, Helobdella stagnalis, Hemiclepsis marginata asiatica, and Theromyzon sp.; Erpobdellidae – Erpobdella octoculata; Hirudinidae – Poecilobdella granulosa; and Haemopidae – Haemopis indicus. Leeches play key roles in prey-predator dynamics and host-parasite relationships in freshwater ecosystems and as bioindicators of water pollution. This study provides essential data for taxonomic accounts and the diversity of leech fauna in Dal Lake.

Hopf bifurcation analysis of a two‐delayed diffusive predator–prey model with spatial memory of prey

In this paper, we consider a diffusive predator–prey model with spatial memory of prey and gestation delay of predator. For the system without delays, we study the stability of the positive equilibrium in the case of diffusion and no diffusion, respectively. For the delayed model without diffusions, the existence of Hopf bifurcation is discussed. Further, we investigate the stability switches of the model with delays and diffusions when two delays change simultaneously by calculating the stability switching curves and obtain the existence of Hopf bifurcation. We also calculate the normal form of Hopf bifurcation to determine the direction of Hopf bifurcation and the stability of bifurcation periodic solutions. Finally, numerical simulations verify the theoretical results.

The dynamic of two prey–One predator food web model with fear and harvesting

This study presents a predator-prey model within a food web, incorporating the impact of fear responses in prey and the effects of harvesting. The analysis begins by exploring equilibrium points and assessing the boundedness and uniqueness of the model's solutions. The local stability of each of the seven possible equilibrium points is then evaluated. To determine global stability, a suitable Lyapunov function is utilized. Finally, numerical simulations are carried out across different parameter values, with graphical representations provided to support the analytical findings.