A geometric analysis of the Bazykin-Berezovskaya predator-prey model with Allee effect in an economic framework

The mountain pine beetle (MPB) exhibits all the hallmarks of a cross-scale disturbance agent, with nonlinear population dynamics driven by positive density-dependent feedbacks that link fast behavioural processes to slow, landscape-level environmental change. Although this feedback loop typically fuels rapid irruption, it can in principle be reversed through sustained population removal. We document such a reversal in Alberta’s commercial pine forest over 2006–2023. No single causal variable effectively captures the collapse mechanism; instead, collapse reflects a syndrome involving two sets of drivers operating at different spatial scales. Intrinsic density-dependent factors include beetle pressure measured at breast-height on the stem, attacks up the height of the stem, and the density of attacked trees in the surrounding cluster. Extrinsic environmental drivers operating at the landscape scale include winter temperatures, drought, pine volumes, and pine ancestry. Despite several million freshly attacked trees detected annually from 2009 to 2012, outbreak expansion did not occur. We explain why the removal of freshly attacked trees is the most plausible explanation for the observed decoupling between r (seasonal recruitment) and R (interannual expansion), which were uncharacteristically uncorrelated throughout 2006–2019. This decoupling indicates that the positive feedback loop linking brood productivity to landscape-level spread was disrupted during the years when it would normally have been strongest. Cold winters in 2018–2020 contributed to the final collapse, but the decisive factor was the earlier disruption of density-dependent feedback through green-tree removal, which limited population growth and tree mortality from 2010 to 2017 and prevented any rebound after 2020. This case illustrates how cross-scale outbreak collapse can be engineered in systems with strong positive feedbacks, and places MPB within a broader class of species whose persistence depends on remaining above population-viability thresholds set by Allee effects. • MPB brood productivity and outbreak spread were decoupled across 2006–2019. • Population removal broke the positive feedback driving beetle irruption. • Cross-scale collapse occurred despite strong density-dependent recruitment. • Cold winters aided collapse, but feedback disruption was the decisive factor. • Five management stages are identified for suppressing irruptive bark beetle systems.

Impacts of hunting cooperation among predators on predator-prey dynamics.

Dynamical Study of a Predator-Prey Model Including Allee Effect, Prey Group Defence, and Extra Food for the Predators

As populations decline under climate change and other stressors, survivors become increasingly isolated. Allee effects in such fragmented populations may arise during reproduction, particularly for sessile organisms such as corals, reducing fertilisation potential and increasing the probability of recruitment failure. However, the critical distances between corals needed to maintain viable populations remains uncertain. We experimentally investigated fertilisation patterns in the broadcast spawning coral Acropora cf. hyacinthus using a manipulated patch, spacing colonies at increasing distances downstream from a central spawning aggregation. Field samples showed a steep decline in fertilisation with increasing intercolonial distance and genetic paternity analysis indicated most parents were located within 3 m of each other. Colony position relative to the current direction also influenced fertilisation success, with 81% of sequenced progeny sired by upstream colonies. Simulations projecting natural population distributions onto a virtual grid, incorporating experimental results, indicated that populations remained well-mixed at typical adult densities. Yet at reduced densities approaching 0.01 colonies m-2, reproductive isolation and patchy breeding units emerged. These findings demonstrate that colony isolation substantially impairs reproductive connectivity, with additional influence from the spatial arrangement of colonies. Conservation strategies should prioritise maintaining colony density and spatial connectivity to support viable coral populations on degraded reefs.

Multiple bifurcations and managing chaos: A discretized ratio-dependent Holling–Tanner predator–prey model with Allee effect in prey

Acceleration or finite speed propagation in integro-differential equations with logarithmic Allee effects

The impact of Allee effect and poaching in eco-epidemic model with gestation delay for predator

In this paper, we extend the classical Hastings–Powell tri-trophic food chain model by incorporating two ecologically realistic mechanisms: predator-induced fear in prey and a weak Allee effect. A rigorous mathematical analysis is carried out, including positivity, boundedness, dissipativeness, and persistence of solutions, followed by a detailed investigation of all equilibrium points and their local and global stability properties. Through bifurcation analysis, we identify that govern system transitions. Sensitivity analysis using Partial Rank Correlation Coefficients (PRCC) highlights the most influential parameters for prey, predator, and top predator dynamics. Numerical simulations, conducted via MATLAB and MATCONT, further reveal complex dynamical regimes. In particular, variation of the prey-handling time parameter [Formula: see text] drives the system into chaos, as confirmed by a positive maximum Lyapunov exponent (MLE). Importantly, we demonstrate that chaos can be suppressed by strengthening the weak Allee effect ([Formula: see text]) and predator-induced fear ([Formula: see text]). Two-parameter bifurcation diagrams for [Formula: see text], [Formula: see text], and [Formula: see text] provide deeper insights into the combined ecological effects of these parameters. The results, supported with ecological interpretations, show how behavioral responses and Allee effects can regulate chaotic oscillations and promote system stability, offering useful theoretical insights for ecological management and conservation.

The emergence of spatial distributions in predator–prey dynamics is a significant topic. In this study, we respond to this problem by creating an intriguing predator–prey model that is time- and space-discrete. The fear effect of predators on prey’s reproduction, Allee effect of predators, and the species’ self-diffusion is considered. The discrete model is represented using a coupled-map lattice, which assumes a nonlinear link between the predator–prey reaction and dispersal stages. The feasible fixed points and their stability are derived along with the conditions of Flip and Neimark–Sacker bifurcations. Fear effect changes the system’s dynamics from stable fixed point to invariant closed curve to stable fixed point again, whereas time step destabilizes the system by formation of an invariant closed curve. The Allee effect stabilizes the system but a further increase in the Allee effect reveals a critical threshold triggering bi-stability and subsequent predator extinction. To discriminate between chaotic and regular behaviors, maximum Lyapunov exponents are depicted. The condition for Turing instability is described and different instability regions are identified. Simulations reveal that spatial movement of both the species increases the spatial heterogeneity and provides a fascinating diversity of spatial patterns, including regular homogeneous, oscillation, quasi-periodic and irregular chaos. In this discrete system, spatiotemporal chaos even arises in the Turing region, and the initial population influences species dispersion. This study shows how the discrete model’s nonlinear dynamics better represent the pattern generation intricacy of predator–prey systems.

ABSTRACT In this paper, memory‐based diffusion is incorporated into a predator‐prey model with a double Allee effect. Turing instability, Hopf bifurcation, double Hopf bifurcation, and Turing‐Hopf bifurcation are analyzed by selecting the memory‐based diffusion coefficient and time delay as bifurcation parameters. Using center manifold theory and the normal form method, the paper derives a general expression for the normal form, which reveals complex spatiotemporal patterns near the Turing‐Hopf singularities. The results show that when the memory‐based diffusion coefficient satisfies certain conditions, Turing instability occurs. Additionally, the memory‐based delay can induce the spatiotemporal patterns of Hopf, double Hopf, and Turing‐Hopf bifurcations. Specifically, the system generates inhomogeneous stable periodic solutions with distinct spatial frequencies. Our results demonstrate that memory‐based diffusion can induce diverse population distribution patterns.

We try to better understand how a spatial barrier may affect the spreading of an invading species via numerical analysis of some variations of a free boundary model in [1, 2] (where only homogeneous environment was considered). Here we incorporate a spatial barrier by replacing a bistable growth term f(u) in the model with f(x,u)=u(r(x)-u)(u-θ), where θ ∈ (0, 1/2) and r(x)=1 except in the barrier region [x0,x0+l], in which r(x) becomes negative away from its boundary, representing the biological assumption that the environment becomes hostile to the species inside the barrier. A parameter α > 0 in the expression of r(x) is used to characterize the severity of the environmental hostility. We find that when all the other parameters are fixed there exists a critical value l* of the barrier length l such that successful spreading is continued past the barrier region when l < l*, and the propagation is blocked when l > l*. Similarly we show numerically that when all the other parameters are fixed, there is a critical value α* of the barrier severity α such that propagation can be continued when α < α*, but it is blocked when α > α*. The dependence of l* (respectively α*) on the other parameters are also analysed. To include temporal fluctuations of the environment, we further replace r(x) by a(t)r(x) with a(t) a positive time-periodic function of average 1, to represent the periodic modulation of the environment. Our numerical simulations suggest that increasing the magnitude of temporal variation enhances the ability of species invasion, while increasing the frequency of such variation reduces this ability. To see how Allee effect may influence the invasion with a barrier, our results based on a bistable f discussed above are compared with that for a model obtained from a standard monostable function (no Allee effect), namely f=u[r(x)-u] with the same r(x). A parallel numerical analysis shows that qualitatively everything is the same in the monostable case, including the numerical results incorporating seasonal changes (with r(x) replaced by a(t)r(x)). However, these numerical simulations indicate that in the bistable case (Allee effect included) the invasion is more likely to cross the barrier than in the monostable case (no Allee effect), suggesting the counter intuitive conclusion that Allee effect may increase the chance of invading across a barrier. Dedicated to Professor Shigui Ruan for his 60th birthday.

This study constructs a discrete-time model featuring the Holling type II functional response to investigate how the Allee effect drives the appearance of both periodic and chaotic behaviors. Through rigorous algebraic analysis, it is demonstrated that Neimark-Sacker (NS) bifurcations occur when varying the Allee parameter n and the prey's intrinsic growth rate r within the positive quadrant. By applying the center manifold theorem and core results from bifurcation theory, a solid theoretical framework is established to track these qualitative changes. Extensive numerical simulations validate that the system undergoes a subcritical NS bifurcation as n and r change, marking a shift from stable equilibria into chaotic oscillations with increasing Allee intensity. Moreover, the model reveals the coexistence of periodic attractors of periods 11 and 34 over certain (n,r) pairs, underscoring how initial population levels critically shape long-term outcomes. A parallel analysis contrasts this Holling II formulation with its Holling type I analog. By varying the Allee threshold n, it becomes apparent that the type I system collapses at a lower Allee value than the type II system. This disparity suggests that predation saturation-a hallmark of Holling II-confers greater resilience under strong Allee effects. Finally, a comparative evaluation of both functional responses' advantages and limitations highlights the need for adaptive management strategies to preserve biodiversity and maintain ecological stability.

Hopf bifurcation and stability in a fractional-order age-structured predator-prey model with Allee effect and dual delays

In this paper, we investigate Turing instability in a diffusive predator-prey system with an opportunistic predator and a weak Allee effect in the prey population. We begin by applying the upper-lower solution method to establish the existence of positive solutions and to derive estimates for both the positive and steady-state solutions. We then show that diffusion can trigger Turing instability in the homogeneous steady states of the system. Numerical simulations are carried out to support the analytical results. In particular, we further examine a stochastic reaction-diffusion version of the model by introducing additive white noise. A comparison between the stochastic and deterministic cases reveals that white noise can enhance the stability of the system.

Рассмотрено влияние сильного эффекта Олли в популяции жертвы на динамику модели «хищник – жертва» с идеальным свободным распределением (ИСР). Приводится сравнение стационарных решений для двух систем: с эффектом Олли и без него. Показано, что жертва всегда сохраняет ИСР, в то время как хищник имеет сложные ИСР–подобные распределения, зависящие от функции ресурса и параметра Олли. Наличие сильного эффекта Олли у жертвы приводит к смене устойчивости тривиального нулевого состояния равновесия и появлению нового неустойчивого решения. При этом диапазон устойчивости стационарного состояния, отвечающего сосуществованию двух видов, уменьшается на величину, пропорциональную па-раметру Олли и обратно пропорциональную ресурсу. Найдено, что изменение параметра Олли напрямую влияет на численность популяции хищника (чем больше параметр, тем меньше численность), в то время как популяция жертвы остается неизменной. Установлено, что с увеличением параметра Олли для некоторых точек ареала (в силу зависимости функции ресурса от пространственной координаты) стационарное решение с двумя видами становится неустойчивым для малых возмущений системы. В результате распределение популяций содержит области, в которых их численность равна нулю. Приведены вычислительные эксперименты, демонстрирующие наличие осциллирующего режима для двух видов. Система без эффекта Олли сохраняет колебания и общий вид пространственного распределения при малых возмущениях начальных условий. При сильном эффекте Олли возможны локальные колебания, вызванные тем, что условие их появления зависит от изменяющейся по пространству функции ресурса.

The Allee effect plays a vital role in species conservation and management in ecological systems. With increasing predation risk, prey individuals are likely to become more fearful and vigilant, which in turn reduces their mate-searching rate. Here, we study a discrete-time predator-prey model in which the inverse of the searching efficiency parameter is replaced by the product of predator density and a proportionality constant representing the intensity of predation risk. We examine the system's stability and bifurcation in detail and find that it undergoes both flip and Neimark-Sacker bifurcations. We also analyse the system's dynamics by simultaneously varying two key parameters to reveal rich and complex behaviours. We observe bistabilities between different pairs of coexisting attractors with intricate basin boundaries. We notice that a decrease in the searching efficiency of mating partners due to fear of predation risk can have stabilizing and destabilizing effects on population dynamics and species persistence.

This paper investigates a spatiotemporal predator-prey model that incorporates the Allee effect, the fear effect, prey-taxis, and harvesting within a Beddington-DeAngelis functional framework. The model captures the combined influence of biological interactions, behavioral responses, and harvesting activities on population dynamics in a spatially heterogeneous environment. The global existence, positivity, and boundedness of classical solutions are first established under appropriate parameter conditions. The existence and local stability of homogeneous steady states are then analyzed, and the conditions for diffusion-driven instability are derived to characterize the onset of spatial patterns. Using weakly nonlinear analysis, amplitude equations are developed to describe the modulation of spatial modes near the bifurcation threshold. Numerical investigations are conducted to complement the theoretical analysis: bifurcation diagrams are employed to examine the effects of biological parameters such as the Allee threshold ($ \beta $), fear intensity ($ \gamma $), and conversion efficiency ($ \varepsilon $), while spatiotemporal simulations are performed to visualize different scenarios and demonstrate the impact of prey-taxis on pattern formation and population organization.

This paper investigates a predator-prey system with the Allee effect under homogeneous Neumann boundary conditions. Firstly, a priori estimates of the positive steady state solutions are obtained by using the maximum principle, and using the energy method, the nonexistence of nonconstant positive steady states is proved. Secondly, the Turing instability of the positive steady state is discussed, and the existence of nonconstant positive steady state solutions is obtained by the Leray-Schauder degree theory. Then, local and global bifurcations at simple eigenvalues are studied by using the bifurcation theory and the bifurcation direction is determined. Additionally, the local bifurcations at double eigenvalues are analyzed by applying the spatial decomposition and the implicit function theorem. Finally, the analytical results of numerical simulations are verified and supplemented.

This study develops a predator–prey model that incorporates logistic growth of the prey population, a Holling type-II functional response, and both an Allee effect and intraspecific competition in the predator population. The nonlinear interactions between species are first examined using a deterministic framework. To better capture real ecological conditions, a stochastic component is then introduced to account for environmental variability and random fluctuations affecting population interactions. A comparative analysis of the deterministic and stochastic models is carried out to investigate the influence of ecological and environmental factors on population dynamics, with particular emphasis on stability, persistence and extinction. Equilibrium points are analyzed to identify conditions that promote species coexistence or lead to population decline. The stochastic model further enhances understanding of predator–prey dynamics by revealing system vulnerability and resilience under environmental uncertainty. Numerical simulations are used to illustrate the theoretical results and to demonstrate the applicability of the model to real-world ecological systems. By integrating deterministic and stochastic perspectives, this study contributes to ecological modeling and provides insights relevant to ecosystem management and biodiversity conservation.