In this letter, we revisit the stochastic model coupling forest growth with human activities developed in one of our previous papers Yu et al. (2025). It was shown there that environmental noise may trigger critical transitions from the high forest cover state to the low cover state. However, the likelihood and time for the occurrence of such tipping events remain not explicitly quantified, which are essential for anticipating ecological degradation and designing sustainable management strategies. This letter addresses this issue by examining the relationship between stochasticity and the probability or time of tipping to the degradation state (the collapse state or the extinction state). Our results indicate that the time of tipping to the degradation state falls as the noise intensity increases; but the probabilities of tipping to the collapse state and the extinction state are different: the tipping probability to the extinction state increases monotonically with noise intensity, whereas the tipping probability to the collapse state exhibits a nonmonotonic dependence on noise intensity. Moreover, we establish a risk classification for different initial conditions, finding that stochasticity exerts a dual effect on human–environment interactions: it may promote forest persistence in regions that would otherwise degrade deterministically, while increasing the degradation risk in otherwise persistent areas. Also, forest-related noise inhibits forest degradation, whereas human-related noise hastens it. These results offer valuable theoretical insights for protecting forest ecosystems, suggesting that precautionary management and early warning strategies can be formulated to avert catastrophic shifts.

In a modified Holling–Tanner predator–prey model, the introduction of a constant prey refuge can lead to complex dynamics, including multi-type bistability and complex bifurcations, which is highly susceptible to environmental perturbations. In this paper, we conduct a comprehensive exploration of the effects of environmental stochasticity on the critical transitions of the model when it exhibits bistability, particularly in conjunction with the capacity of the prey refuge, using the stochastic sensitivity function (SSF) technique. Specifically, we focus on two types of bistability: one between two positive equilibria and the other between an interior equilibrium and a limit cycle surrounding it. The critical values of noise intensity and prey refuge for the occurrence of regime shifts are identified. If we further consider stable coexistence as our target, the parameter ranges for the prey refuge and noise intensity are also determined. In particular, we conclude that a larger value of the prey refuge parameter is beneficial while a moderate value of noise intensity is more advantageous for this target.

. Alien mussels can seriously alter the freshwater ecosystems in which they invade, both biotically and physically. Their invasions are inevitably influenced by random changes in the environment. To evaluate the impact of environmental stochasticity on the dynamics of mussel invasion, we propose and study a stochastic dynamical model that describes the interactions between algae, mussels, and sediments, simultaneously incorporating the effects of environmental white noise and state switching. A comprehensive analysis of the asymptotic behavior of the model is performed. It is shown that under some mild conditions, the long-time behaviors of the model can be characterized by a critical value lambda: If lambda < 0, the invasive mussels will die out exponentially; while if lambda > 0, the model possesses an invariant probability measure, and the transition probability of the solution process converges to that of the invariant measure. The rate of convergence is also obtained. Numerical examples are given to illustrate our theoretical research findings.

Human activities can influence the state of ecosystems, and the consequent variation in ecosystem services in turn affects people’s perceptions and behaviors, thus forming a feedback loop. In this paper, a novel coupled human-environment model is proposed and analyzed. The model employs a replicator dynamics equation to describe the human behavioral decision-making process and couples it to an ecological subsystem representing lake eutrophication. Analysis of the model shows that it has richer dynamical behaviors, including multistability and sustained periodic oscillations. Interestingly, for the scenario when the model possesses tristability, we find that increasing the cost of conservation too quickly can trigger rate-induced tipping, causing the system to switch between oligotrophic (or eutrophic) and intermediate steady states. Furthermore, there is a threshold for the rate of change in the cost of conservation below which the probability of tipping from an oligotrophic state to an intermediate nutrient state is always greater than that of switching from an intermediate nutrient state to a eutrophic state. Above this threshold, however, the probability of tipping from the intermediate state to the eutrophic state prevails. Our results suggest that limiting only the magnitude of lake conservation costs is not sufficient to control lake deterioration and that the rate of increase in conservation costs should also be considered.

In this paper, inspired by the forage-maturation hypothesis, we propose and investigate a novel resource-consumer model that incorporates forage quality, represented by digestibility, which declines exponentially with increasing biomass. Our aim is to reveal the coupled effects of resource biomass and quality on system dynamics. To this end, we first analyze the existence and stability of equilibria as well as the possible bifurcations the model undergoes, and then perform some numerical simulations to illustrate and complement the obtained theoretical results. Compared to the traditional resource-consumer model with no consideration of resource quality, our model can exhibit much more complex dynamics, including rich bifurcation phenomena such as Hopf, transcritical, saddle-node, homoclinic as well as cusp and Bogdanov-Takens bifurcations and two types of bistability. Both theoretical and numerical findings indicate that large resource biomass with low quality can lead to the extinction of consumers; however, for a herbivore population with low mortality rate, a moderate level of forage quality is more conducive to its persistence. These can provide fresh insights into maintaining the ecological balance within forage-herbivore systems.

The spatial distribution of oyster reefs is an important indicator for assessing environmental changes in nearshore fishery habitats. However, due to tidal fluctuations, images of oyster reef distribution acquired under low-light conditions such as early morning or evening often exhibit common issues such as bright spots and shadows. Thermal infrared (TIR) images, which are unaffected by external lighting conditions, can effectively address this problem. Aerial imaging of Liya Mountain, Haimen, Jiangsu Province, China, was conducted in this study. Based on unmanned aerial vehicles (UAVs) imagery acquired in 2025 using multispectral and TIR sensors, the total oyster reef area was estimated to be 6.61 ha. When compared with the oyster reef distribution derived from visible light aerial imagery collected in 2023 under favorable environmental conditions, this represents a decrease of 0.36 ha (5.4%), with the largest individual reef measuring 3388.17 m2. To demonstrate the improvement in extraction accuracy achieved by integrating TIR data with multispectral imagery, the research team compared the extraction accuracy for oyster reefs of different sizes: a 1.91% improvement was observed for small reefs, a 9.02% improvement for middle reefs, and an 18.98% improvement for large reefs. Experimentally, the emissivity of oyster reefs was determined to be 0.982 ± 0.002 using an isothermal method in the laboratory. The emissivity derived from in situ measurements showed similar values, supporting the reliability of the laboratory result and providing a crucial parameter for the inversion of reef surface temperature. Experimental results demonstrate that the TIR band can effectively enhance the spatial accuracy of oyster reef measurements under low-light conditions.

Spatial memory is essential for regulating the spatiotemporal distribution of plankton. In this paper, a spatial memory term is incorporated into a diffusive toxin-producing phytoplankton-zooplankton model with three-dimensional patches. The aim is to examine the combined influence of memory delay and the resulting cross-diffusion coefficient on the spatiotemporal dynamics of the populations. We first prove the existence and uniqueness of solutions for the model and analyze the existence and stability of its homogeneous equilibria. Next, for the model without memory delay, we take the cross-diffusion coefficient as the bifurcation parameter and derive the conditions under which Turing instability arises. When the delay effect is included, the model undergoes an n-mode Hopf bifurcation under the joint action of cross-diffusion and delay. Finally, we support the theoretical analysis with numerical simulations. The results show that toxins play a substantial regulatory role in sustaining ecosystem stability. In addition, the introduction of cross-diffusion leads to spatially heterogeneous steady-state distributions. The inclusion of memory delay further causes the model to exhibit periodic temporal oscillations, giving rise to spatiotemporally coupled non-uniform patterns. Biologically, toxin concentration and spatial memory cycles help promote the coexistence of phytoplankton and zooplankton, thus acting as a key mechanism for preserving ecosystem homeostasis.

Understanding the dynamics of the predator and prey populations is essential for maintaining ecosystem stability and biodiversity, both in theory and in practice. Because generalist predators consume multiple prey species, they can remain in the ecosystem even when certain prey are absent. In this work, we formulate a stochastic generalist predator-prey model incorporating a sigmoidal functional response. We first prove the global existence and uniqueness of solutions, as well as the boundedness of positive solutions. We then derive sufficient conditions for the persistence and extinction of both the predator and the prey populations. Using the stochastic sensitivity functions (SSF) method, we construct confidence ellipses for the random state and identify the critical noise intensity at which steady-state transitions occur. Finally, we investigate the mean first passage time (MFPT) and probability of the system's extinction. Our results show that higher noise intensity accelerates the extinction process and increases the likelihood of extinction. We further find that increasing predator conversion efficiency makes it less probable for the predator population to drop to a low level and reduces the chance of simultaneous collapse of both prey and predator populations. In particular, noise has a stronger impact on prey than on generalist predators.

In this letter, we revisit a Lesile-Gower intraguild predation model proposed by Safuan et al. in the paper Safuan et al. (2013). It was shown there as the biotic resource enrichment parameter gamma varies, the model can undergo a transcritical bifurcation which might explain two alternative scenarios: one is the coexistence of three populations, and the other is the extinction of the intra prey. That is, the survival of intra prey population is solely determined by the biotic resource enrichment. In fact, using the same parameter gamma as bifurcation parameter, the model can also undergo a saddle-node bifurcation at some critical value gamma SN, which might explain another two alternative scenarios: one is the bistability between a positive equilibrium and an intra prey extinction one, and the other is the extinction of the intra prey. This means that the intra prey species may undergo a catastrophic shift when gamma increases passing through gamma SN. This is established by proving the existence of positive equilibria and determining their stability, theoretically and numerically.

In this study, a numerical model consisting of high-resolution hydrodynamic and Lagrangian particle tracking modules based on the Finite-Volume Coastal Ocean Model framework was established to simulate the hydrodynamic conditions and characteristics of the sedimentation of aquaculture-derived organic matter (AOM) from cage aquaculture in Sansha Bay. The results showed that Sansha Bay was characterized by regular semidiurnal tides and large tidal ranges. Reciprocating currents with main currents directed northward and southward during the rising and falling tides, respectively, predominated the main channels of the bay. Residual feed had larger settling velocities than feces. The maximal dispersion distances of residual feed and feces during the spring tide were 217.1 and 1805.7 m, respectively, three times those during the neap tide (74.2 and 675.6 m, respectively). During the spring tide, the largest dispersion distance of AOM occurred at the rush moment. The AOM movement trajectories were mainly controlled by the main currents. Both the tidal structure and current characteristics affected the AOM sedimentation in Sansha Bay. The sedimentation characteristics of AOM were unrelated to feeding intensity. The results of simulations agreed with the field observations in this study, suggesting that the estimated model had a good accuracy and sensitivity.

Mitigating carbon dioxide (CO2) emissions associated with energy generation is crucial for addressing the climate crisis. To better understand the dynamic relationship between CO2 concentration, human population, and energy consumption in a stochastic environment, we propose and investigate a stochastic carbon emissions model and further consider its near-optimal control (NOC) problem. We first focus on the natural evolution scenario without intervention measures to analyze the dynamic behavior of the carbon emissions system under environmental fluctuations. The results suggest that when environment noise is sufficiently large (such that ϕ<0), it will lead the population to collapse, thereby reducing energy consumption to zero, and eventually returning CO2 concentration to pre-industrial level. This is an unsustainable scenario ecologically for the model. When environment noise is not too large (such that ϖ>0), there exists a unique ergodic stationary distribution. To effectively reduce the CO2 concentration while ensuring a reasonable population size, we then develop a NOC system that incorporates two intervention strategies. Using the Pontryagin stochastic maximum principle, we establish necessary and sufficient conditions for the existence of the near-optimality. Theoretical and numerical results demonstrate that effective CO2 mitigation strategies must consider both ecological sustainability and economic feasibility. From the perspective of policymakers, this study emphasizes the importance of dynamically adjusting emission reduction strategies across different development stages. Such adaptive decision-making can effectively alleviate atmospheric CO2 concentration while ensuring economic and ecological sustainability.

The dynamic behavior of alien mussels interacting with algae after arriving in a new environment has long been a focus of invasion ecology research. This paper extends and analyzes a classical mussel-algae model by incorporating a time delay in mussel filter feeding and accounting for environmental variability. We theoretically study the stochastic dynamics, including the global existence and uniqueness of the positive solution, the existence of a unique stationary distribution, and mussel extinction, using tools from stochastic analysis. Furthermore, we derive an explicit expression for the probability density function around the quasi-stable equilibrium by solving the corresponding Fokker-Planck equation. Our theoretical and numerical results indicate that: (a) larger environmental disturbances or artificial removal can effectively prevent the survival of alien mussels in novel habitats, (b) a decreased filter feeding rate leads to an accelerated extinction rate of mussels, and (c) an increased consumption constant c decelerates the transition rate of mussels from the initial state to the extinction state, as analyzed through the mean first passage time of mussels. These findings highlight the complex interaction between intrinsic and extrinsic factors in influencing the invasion dynamics of alien mussels.

Human and environmental systems should not be viewed in isolation from each other but as a complex integrated system since humans not only influence ecosystem services and functions but also respond to changes in the ecosystem. Additionally, stochastic perturbations play a crucial role in natural systems, and stochastic factors associated with social and ecological systems can significantly affect the dynamics of coupled models, such as noise-induced tipping. In this paper, we propose a coupled human-environment model with noisy disturbances that includes the dynamics of forest conservation opinions within a population and the natural expansion and harvesting of forest ecosystems. We investigate how stochasticity triggers critical transitions between high and low forest cover states (or a stable oscillatory state) using social and ecological fitting parameters from old-growth forests in Oregon. Based on landscape-flow theory from non-equilibrium statistical mechanics, we quantify the global stability and robustness of equilibria and limit cycles using barrier height and average flux. We find that the stability of the high forest cover state weakens, and the low forest cover state becomes increasingly stable as noise intensity increases. Conversely, an increase in the intensity of injunctive social norms favors the global stability of the high forest cover state. Moreover, only a sufficiently small forest protection cost will allow forest cover to be maintained at a high level. Finally, a sensitivity analysis of the parameters of the coupled system is conducted, revealing the key factors affecting the global stability and critical transitions of high and low forest cover states.

Monitoring Larimichthys crocea aquaculture in a low-cost, efficient and flexible manner with remote sensing data is crucial for the optimal management and the sustainable development of aquaculture industry and aquaculture industry intelligent fisheries. An innovative automated framework, based on the Segment Anything Model (SAM) and multi-source high-resolution remote sensing image data, is proposed for high-precision aquaculture facility extraction and overcomes the problems of low efficiency and limited accuracy in traditional manual inspection methods. The research method includes systematic optimization of SAM segmentation parameters for different data sources and strict evaluation of model performance at multiple spatial resolutions. Additionally, the impact of different spectral band combinations on the segmentation effect is systematically analyzed. Experimental results demonstrate a significant correlation between resolution and accuracy, with UAV-derived imagery achieving exceptional segmentation accuracy (97.71%), followed by Jilin-1 (91.64%) and Sentinel-2 (72.93%) data. Notably, the NIR-Blue-Red band combination exhibited superior performance in delineating aquaculture infrastructure, suggesting its optimal utility for such applications. A robust and scalable solution for automatically extracting facilities is established, which offers significant insights for extending SAM’s capabilities to broader remote sensing applications within marine resource assessment domains.

Recreational fisheries, often recognized as fishing activities undertaken by individuals for sport and leisure, can be conceptualized as integrated systems where changes in human activities have significant implications for the interconnected components of the fishery ecosystems. These systems often exhibit nonlinear dynamics and are susceptible to abrupt and irreversible shifts in function and structure, referred to as critical transitions. Most existing studies on critical transitions in fishery ecosystems focus on the magnitude of changes in fishing practices. In this study, we examine a recreational fishery model that incorporates the prey and its predators, both of which are subject to fishing by anglers. Our analysis demonstrates that the fishery ecosystem is sensitive not only to the magnitude of changes in fishing pressure but also to the rate at which these changes are implemented. Surprisingly, for fixed fishing pressure of predators, a slow increase in fishing pressure of the prey, even when aiming at a relatively high fishing target, can sustain the prey fish stock at a sustainable biomass level, while a fast increase in fishing pressure can lead to the collapse of prey population. Additionally, we identify critical thresholds for the rate of increase in fishing pressure for different initial population sizes and accordingly assign warning levels to these values. Furthermore, to evaluate the risk of rate-induced tipping, we introduce two key notions of tipping time and rate-tipping probability. Interestingly, we find that as the rate of change of fishing pressure (\lambda ) increases, the tipping probability first increases rapidly, then slowly, and finally saturates, while tipping time decreases gradually, that is, the increase of \lambda accelerates the collapse of prey population. The scenario in which the fishing pressures of the prey and predators vary synchronously is also considered. Our study offers valuable insights for fisheries management decision-making, not only emphasizing the importance of considering the magnitude of management adjustments but also highlighting the temporal dynamics of implementing these changes to achieve desired outcomes.

In this paper, we present the Euler equation of steady periodic equatorial water waves in two-layer flows with different densities and generalise the two Stokes’ definitions for the velocity of the wave propagation. We further demonstrate that the excess potential energy density of nonlinear equatorial two-layer waves is always positive, while the excess kinetic energy density is negative.

Larval type dimorphism, a special strategy utilized by some marine invertebrates to reproduce, has been commonly recognized in a clade of herbivorous sea slugs, the Sacoglossa. To make clear the underlying evolution mechanism and inherent laws of reproductive strategy in these marine invertebrates, in this paper, we propose and investigate a stage structure model describing the larval type dimorphism for sacoglossan sea slugs. We first perform a global analysis of the model when sea slugs have a mixed reproductive strategy. It is shown that the model's dynamics is completely determined by the ecological reproductive index of sea slugs R-0: When R-0>1, both the two type larvas can coexist with the adults; while when R-0 <= 1, the total sea slug species will go extinct. Then taking the adults' benthic offspring reproduction rate q as a trait, we proceed to investigate the adaptive evolution of sea slugs' reproductive strategy. Our theoretical and numerical results indicate that a mixed reproductive strategy could evolve to a simple one under certain circumstances, and vice versa, and the coexistence of two different mixed reproductive strategies is impossible. In addition, we also consider the impacts of environmental fluctuations on the dynamics of sea slug population, finding that environmental noises potentially affect the diversity of sea slug species. Our results partially explain the mystery in marine evolutionary ecology regarding why so few invertebrates exhibit reproductive dimorphisms resulting in alternative larval types.

To address whether alien mussels can successfully invade a new habitat, we investigate the dynamic behaviors of the mussel-algae interaction using both deterministic and stochastic models of differential equations by simultaneously incorporating the manual removal of mussels. In the deterministic model, we derive the ecological invasion index of mussels, denoted as R-0. We find that when R-0 < 1, the invasion of the mussel population may fail, whereas when R-0> 1, the invasion is successful. Additionally, we observe a backward bifurcation phenomenon, where the success of the invasion also depends on the initial population size of mussels; sufficiently large initial populations can lead to successful invasions even when R-0< 1. Similarly, in parallel to the deterministic model, the stochastic invasion index (R-0(s)) of mussels serves to distinguish model outcomes. When R-0(s) < 1, it is highly likely that the mussel invasion fails, while when R-0(s) > 1, the invasion is successful, facilitated by the presence of an ergodic stationary distribution. Theoretical analysis demonstrates that the rate at which all solutions converge to the stationary distribution is polynomial. Moreover, we conclude that human intervention represents the most effective approach to mitigate invasion, and it is possible to sustain both mussel and algae populations through proper mussel removal strategies.

Human activities and climate change are severely destabilizing fishery ecosystems globally, leading to a significant reduction in biodiversity, adversely affecting the global fishery market, and resulting in substantial economic losses. To better understand the intrinsic mechanism of interaction between fishery ecology and fishery market in a stochastic environment, this paper presents a stochastic fishery-economy coupled model. We establish sufficient conditions for the persistence and collapse of both the fishery resource and the fishery market, detailing three scenarios: (i) the fishery market collapses while the fishery resource remains; (ii) both the fishery resource and the market fail, and (iii) both the fishery resource and the market persist. Additionally, we estimate the probability of fishery resource collapse within a given time. To elucidate the intrinsic mechanisms by which market price and stochastic disturbances affect the fishery resource and the market, we derive the explicit local probability density of the stochastic model using Fokker-Planck equation. Our findings suggest that decision-makers can mitigate the negative impacts of stochastic environments on the fishery resource and the market by adjusting the price parameter. Interestingly, our results reveal that environmental stochasticity acts as a double-edged sword: while noise can make the fishery market vulnerable and reduce fishing pressure on resources, thereby promoting fishery resource growth, excessive noise can also directly inhibit the growth of fishery resources. Finally, we illustrate our results through numerical simulations.

In this paper, we explored a modified Leslie-Gower predator-prey model incorporating a fear effect and multiple delays. We analyzed the existence and local stability of each potential equilibrium. Furthermore, we investigated the presence of periodic solutions via Hopf bifurcation bifurcated from the positive equilibrium with respect to both delays. By utilizing the normal form theory and the center manifold theorem, we investigated the direction and stability of these periodic solutions. Our theoretical findings were validated through numerical simulations, which demonstrated that the fear delay could trigger a stability shift at the positive equilibrium. Additionally, we observed that an increase in fear intensity or the presence of substitute prey reinforces the stability of the positive equilibrium.