Plant-herbivore Model Research Articles

Optimized Parameter Estimation and Integrating Neural Network Forecasting of Dynamic Plant-Livestock Model for Early Warning in Agro-Environment Control Systems

The research utilizes the Lotka-Volterra prey-predator model to study Plant-Herbivore dynamics, focusing on the relationship between traditional livestock farming and vegetation conditions. Advanced methods are developed to improve the precision and efficiency of parameter estimation in these models. Neural networks are incorporated to enhance forecasting abilities, and an extension of the Plant-Herbivore models includes Botswana's climate and livestock variables. Efficient parameter space exploration is achieved using the Runge-Kutta method along with Multistart and the local solver $fmincon$ in MATLAB. This method improves parameter estimation accuracy. To address the impact of homogeneity assumptions in the data, estimate aggregation through weighting and time conversion is applied. Furthermore, the study investigates the use of nonlinear least squares to further refine the process, allowing for the identification of parameters that best fit observed livestock data, even with non-linearity. By using optimized parameter estimation techniques along with normalized nonlinear least squares, the cumulative error was reduced from an initial 1563.4521 to a final value of 0.0038, well within the specified thresholds (1.0, 0.1, and 0.01). Comparisons between Autoregressive Integrated Moving Average (ARIMA) and Neural Network Auto-Regressive (NNAR) models showed that NNAR models outperformed ARIMA models, with lower variance estimates (0.000004 - 0.000562) compared to ARIMA (0.103 - 0.155). NNAR models displayed Mean Error (ME) values ranging from -0.0012 to 0.0140, indicating a close match between forecasts and actual values with minor deviations. As a result, NNAR forecasting was used for predicting soil moisture, death, and harvest rates, which were integrated into the extended Plant-Herbivore model. This integration enabled the estimation of livestock production trajectories for 2021-2022, along with corresponding interpretations. The study also assessed the uncertainty propagation from NNAR forecasts onto the Plant-Herbivore dynamic model, revealing an increase in uncertainty with longer lead times.

Dynamics and control of a plant-herbivore model incorporating Allee's effect

This research focuses on the interaction between the grape borer and grapevine using a discrete-time plant-herbivore model with Allee's effect. We specifically investigate a model that incorporates a strong predator functional response to better understand the system's qualitative behavior at positive equilibrium points. In the present study, we explore the topological classifications at fixed points, stability analysis, Neimark-Sacker, Transcritical bifurcation and State feedback control in the two-dimensional discrete-time plant-herbivore model. It is proved that for all involved parameters ς1,ϱ1,γ1 and ϒ1, discrete-time plant-herbivore model has boundary and interior fixed points: c1=(0,0), c2=(ς1−1ϱ1,0) and c3=(ϒ1(1−γ1)2γ1−1,γ1(2ς1+ϱ1ϒ1−2)−ϱ1ϒ1+1−ς12γ1−1) respectively. Then by linear stability theory, local dynamics with different topological classifications are investigated at fixed points: c1=(0,0), c2=(ς1−1ϱ1,0) and c3=(ϒ1(1−γ1)2γ1−1,γ1(2ς1+ϱ1ϒ1−2)−ϱ1ϒ1+1−ς12γ1−1). Our investigation uncovers that the boundary equilibrium c2=(ς1−1ϱ1,0) experiences a transcritical bifurcation, whereas the unique positive steady-state c3=(ϒ1(1−γ1)2γ1−1,γ1(2ς1+ϱ1ϒ1−2)−ϱ1ϒ1+1−ς12γ1−1) of the discrete-time plant-herbivore model undergoes a Neimark-Sacker bifurcation. To address the periodic fluctuations in grapevine population density and other unpredictable behaviors observed in the model, we propose implementing state feedback chaos control. To support our theoretical findings, we provide comprehensive numerical simulations, phase portraits, dynamics diagrams, and a graph of the maximum Lyapunov exponent. These visual representations enhance the clarity of our research outcomes and further validate the effectiveness of the chaos control approach.

Food-limited plant–herbivore model: Bifurcations, persistence, and stability

This research paper delves into the two-dimensional discrete plant–herbivore model. In this model, herbivores are food-limited and affect the plants’ density in their environment. Our analysis reveals that this system has equilibrium points of extinction, exclusion, and coexistence. We analyze the behavior of solutions near these points and prove that the extinction and exclusion equilibrium points are globally asymptotically stable in certain parameter regions. At the boundary equilibrium, we prove the existence of transcritical and period-doubling bifurcations with stable two-cycle. Transcritical bifurcation occurs when the plant’s maximum growth rate or food-limited parameter reaches a specific boundary. This boundary serves as an invasion boundary for populations of plants or herbivores. At the interior equilibrium, we prove the occurrence of transcritical, Neimark–Sacker, and period-doubling bifurcations with an unstable two-cycle. Our research also establishes that the system is persistent in certain regions of the first quadrant. We demonstrate that the local asymptotic stability of the interior equilibrium does not guarantee the system’s persistence. Bistability exists between boundary attractors (logistic dynamics) and interior equilibrium for specific parameters’ regions. We conclude that changes to the food-limitation parameter can significantly alter the system’s dynamic behavior. To validate our theoretical findings, we conduct numerical simulations.

The Study of Stability Analysis of Modified Leslie-Gower Herbivore Model with Allee Effect in Plants

A modified Leslie-Gower plant-herbivore model is studied under the Allee effect. Holling-type II functional response is used to modify the model. Delay differential equations play an essential role in making this model more realistic and complicated. The non-trivial equilibrium E* (P* ≠ 0, H* ≠ 0) of the proposed model is calculated. Moreover, the stability and instability of the state variables, which include plant population P and herbivore population H are described graphically. It is shown that the system represents absolute stability when it has no time parameter (τ). When the time parameter is less than the threshold value, then the system exhibits asymptotic stability. In addition, the system surrenders its stability, and Hopf-bifurcation occurs when the time parameter surpasses the threshold value. The timeseries graphs are also represented. It is demonstrated that the system becomes more stable with the maximum rate of predation. MATLAB software is used to perform the graphs to justify the theoretical results.

Stability and bifurcation analysis of a discrete-time plant-herbivore model with harvesting effect

<abstract><p>The dynamics of plant-herbivore interactions are essential for understanding ecosystem stability and resilience. This article investigated the effects of incorporating a harvesting effect on the dynamics of a discrete-time plant-herbivore system. An analysis was performed to determine the existence and stability of fixed points. In addition, studies have shown that the system experienced transcritical, period-doubling, and Neimark-Sacker bifurcations. Moreover, we provided numerical simulations to substantiate our theoretical results. Our research indicated that harvesting in excessive amounts may have negative effects on the populations of both plants and herbivores. However, when harvesting was done at moderate levels, it promoted the coexistence and stability of both populations. The findings of our analysis provided a deep understanding of the intricate dynamics of ecological systems and underscored the need to use sustainable harvesting methods for the management and preservation of ecosystems.</p></abstract>

Qualitative behavior of a two-dimensional discrete-time plant-herbivore model

Qualitative behavior of a two-dimensional discrete-time plant-herbivore model

Dynamical analysis of a discrete-time plant–herbivore model

In this paper, a discrete-time model of a plant–herbivore system is qualitatively analyzed using difference equations to describe population dynamics over time. The goal is to examine how the model behaves under varying parameter values and initial conditions. Results reveal that the model exhibits diverse dynamical behaviors such as stable equilibria, period-doubling cascade, and chaotic attractors. The analysis indicates that changes in crucial parameters greatly affect the system’s dynamics. This study offers crucial insights into plant–herbivore systems and highlights the value of qualitative analysis in comprehending intricate ecological systems.

Dynamics of a Plant-Herbivore Model Subject to Allee Effects with Logistic Growth of Plant Biomass

This paper examines the relationship between herbivores and plants with a strong Allee effect. When the plant reaches a particular size, the herbivore attacks it. We use the logistic equation to model plant growth and analyze its behavior without herbivores before investigating their interactions. Our study investigates the equilibrium points and their stability, discovering that different fixed points can become unstable due to various bifurcations such as transcritical, saddle-node, period-doubling, and Neimark–Sacker bifurcations. We have identified the Allee threshold, which, if exceeded, can cause both populations to become extinct below that level. However, we have discovered a coexistence equilibrium that is locally asymptotically stable for a range of parameter values above that threshold. Our additional numerical simulations suggest that this area of stability can be expanded. Our results indicate that this system is highly responsive to its parameters. We compare our findings to those of a system without strong Allee effects and conduct numerical simulations to verify our results. By including the Allee effect in the plant population, we enrich the local and global dynamics of the system.

Opposite roles of MAPKKK17 and MAPKKK21 against Tetranychus urticae in Arabidopsis.

After recognizing a biotic stress, plants activate signalling pathways to fight against the attack. Typically, these signalling pathways involve the activation of phosphorylation cascades mediated by Mitogen-Activated Protein Kinases (MAPKs). In the Arabidopsis thaliana-Tetranychus urticae plant-herbivore model, several Arabidopsis MAP kinases are induced by the mite attack. In this study, we demonstrate the participation of the MEKK-like kinases MAPKKK17 and MAPKKK21. Leaf damage caused by the mite was assessed using T-DNA insertion lines. Differential levels of damage were found when the expression of MAPKKK17 was increased or reduced. In contrast, reduced expression of MAPKKK21 resulted in less damage caused by the mite. Whereas the expression of several genes associated with hormonal responses did not suffer significant variations in the T-DNA insertion lines, the expression of one of these kinases depends on the expression of the other one. In addition, MAPKKK17 and MAPKKK21 are coexpressed with different sets of genes and encode proteins with low similarity in the C-terminal region. Overall, our results demonstrate that MAPKKK17 and MAPKKK21 have opposite roles. MAPKKK17 and MAPKKK21 act as positive and negative regulators, respectively, on the plant response. The induction of MAPKKK17 and MAPKKK21 after mite infestation would be integrated into the bulk of signalling pathways activated to balance the response of the plant to a biotic stress.

NEIMARK-SACKER BIFURCATION AND CONTROL OF CHAOTIC BEHAVIOR IN A DISCRETE-TIME PLANT-HERBIVORE SYSTEM

In this study, the dynamics of a discrete-time plant-herbivore model obtained using the forward Euler method are discussed. The existence of fixed points is investigated. A topological classification is made to examine the behavior of the positive fixed point where the plant and the herbivore coexist. In addition, the existence conditions and direction of Neimark-Sacker bifurcation of the model are investigated using bifurcation theory. Hybrid control method is applied to control the chaos caused by Neimark-Sacker bifurcation. Examples including time series figures, bifurcation figures, phase portraits and maximum Lyapunov exponent are provided to support our theoretical results.

On the qualitative study of a two-trophic plant-herbivore model.

The coexistence of plant-herbivore populations in an ecological system is a fundamental topic of research in mathematical ecology. Plant-herbivore interactions are often described by using discrete-time models in the case of non-overlapping generations: such generations have some specific time interval of life and their old generations are replaced by new generations after some regular interval of time. Keeping in mind the dynamical reliability of continuous-time models we presented two discrete-time plant-herbivore models. Mainly, by applying Euler's forward method a discrete-time plant-herbivore model is obtained from a continuous-time plant-herbivore model. In addition, a dynamically consistent discrete-time plant-herbivore model is obtained by applying a nonstandard difference scheme. Moreover, local stability is discussed and the existence of bifurcation about positive equilibrium is shown under some mathematical conditions. To control bifurcation and chaos, a modified hybrid technique is developed. Finally, to support our theocratical results and to show the dynamical reliability of the nonstandard difference scheme some numerical examples are provided.

Perfect Compensation Is Sufficient to Explain Insect Outbreaks Previously Attributed to Overcompensation : (A Comment on Stieha et al., "The Effects of Plant Compensatory Regrowth and Induced Resistance on Herbivore Population Dynamics").

AbstractIn "The Effects of Plant Compensatory Regrowth and Induced Resistance on Herbivore Population Dynamics," which appeared in The American Naturalist in 2016, Stieha et al. argued that overcompensatory regrowth of plant tissues lost to herbivory ("overcompensation") promotes cyclic herbivore outbreaks. In contrast, they concluded that partial regrowth ("tolerance") stabilizes herbivore dynamics, preventing outbreaks. These conclusions were based on a comparison between two plant-herbivore models that differed in two properties: (1) whether biomass could ever be higher after herbivory and regrowth than before herbivory (i.e., is overcompensatory regrowth possible?) and (2) how much herbivory the plants could withstand before only being able to partially compensate for losses (for overcompensating plants, there was a threshold herbivory level above which this occurred, whereas tolerant plants always showed partial compensation). While Stieha et al. supposed that difference 1 was responsible for the increased propensity for outbreaks in their overcompensation model, we show here that, in fact, difference 2 is responsible. Thus, we conclude that Stieha et al.'s results about "overcompensating" plants apply more broadly: the risk of herbivore outbreaks is elevated whenever plants with low-enough herbivore loads can perfectly compensate or overcompensate for losses to herbivory.

Bifurcations, Permanence and Local Behavior of the Plant-Herbivore Model with Logistic Growth of Plant Biomass

Bifurcations, Permanence and Local Behavior of the Plant-Herbivore Model with Logistic Growth of Plant Biomass

Optimal control of a discrete-time plant-herbivore/pest model with bistability in fluctuating environments.

Motivated by regulating/eliminating the population of herbivorous pests, we investigate a discrete-time plant-herbivore model with two different constant control strategies (removal versus reduction), and formulate the corresponding optimal control problems when its dynamics exhibits varied types of bi-stability and fluctuating environments. We provide basic analysis and identify the critical factors to characterize the optimal controls and the corresponding plant-herbivore dynamics such as the control upper bound (the effectiveness level of the implementation of control measures) and the initial conditions of the plant and herbivore. Our results show that optimal control could be easier when the model has simple dynamics such as stable equilibrium dynamics under constant environment or the model exhibits chaotic dynamics under fluctuating environments. Due to bistability, initial conditions are important for optimal controls. Regardless of with or without fluctuating environments, initial conditions taken from the near the boundary makes optimal control easier. In general, the pest is hard to be eliminated when the control upper bound is not large enough. However, as the control upper bound is increased or the initial conditions are chosen from near the boundary of the basin of attractions, the pest can be manageable regardless of the fluctuating environments.

Dynamics of a plant-herbivore model with a chemically-mediated numerical response

A system of two ordinary differential equations is proposed to model chemically-mediated interactions between plants and herbivores by incorporating a toxin-modified numerical response. This numerical response accounts for the reduction in the herbivore's growth and reproduction due to chemical defenses from plants. It is shown that the system exhibits very rich dynamics including saddle-node bifurcations, Hopf bifurcations, homoclinic bifurcations and co-dimension 2 bifurcations. Numerical simulations are presented to illustrate the occurrence of multitype bistability, limit cycles, homoclinic orbits and heteroclinic orbits. We also discuss the ecological implications of the resulting dynamics.

Spatiotemporal patterns of a diffusive plant–herbivore model with toxin-determined functional responses: Multiple bifurcations

In this paper, a homogeneous diffusive plant–herbivore model with toxin-determined functional response subject to the homogeneous Neumann boundary condition in the one dimensional spatial open bounded domain is considered. By using Hopf bifurcation theorem and steady state bifurcation theorem due to Yi et al. (2009), we are able to show the existence of Hopf bifurcating periodic solutions (spatially homogeneous and non-homogeneous) and bifurcating non-constant steady state solutions. In particular, under certain conditions, the globally asymptotic stability of the positive constant steady state solutions and the non-existence of non-constant positive steady state solutions are investigated. These results allow the clear understanding of the mechanisms of the spatiotemporal pattern formations of this ecology model. In particular, our numerical results authenticate that toxicant parameter will play important roles in the stability and instability of the periodic solutions.

带有毒素效应功能反应的植物–食植动物模型的脉冲控制研究

带有毒素效应功能反应的植物–食植动物模型的脉冲控制研究

Impact of plant defense level variability on specialist and generalist herbivores

Most organisms are defended against others, and defenses such as secondary metabolites in plants vary across species, individuals, and subindividual organs. Plant leaves show an impressive variability in quantitative defense levels, even within the same individual. Such variation might mirror physiological constraints or represent an evolved trait. One important hypothesis for the prevalence of defense variability is that it reduces herbivory due to non-linear averaging (Jensen’s inequality). In this study, we explore the conditions under which this hypothesis is valid and how it depends on the degree of specialization of the herbivores. We thus distinguish between generalists, non-sequestering specialists, and sequestering specialists that are able to convert consumed plant defense into own defense against predators. We propose a plant-herbivore model that takes into account herbivore preference, predation pressure on the herbivores, and the three herbivore specialization strategies we consider. Our computer simulations reveal that defense level variability reduces herbivory by all three populations when nutrient concentration is strongly correlated with defense level. If the nutrient concentration is the same in all leaves, the plant benefits from high defense level variability only when the herbivores are specialists that show a considerable degree of preference for leaves on which they perform best.

Stochastic plant-herbivore interaction model with Allee effect.

Environmental noises often affect population dynamics, and hence many benefits are gained in using stochastic models since real life is full of stochasticity and randomness. In this paper a stochastic extension of a model by Asfaw et al. (Int J Biomath 11:1850057, 2018) is considered. Due to the non-linearity of the model, first, a simplified stochastic plant-herbivore model is formulated and analyzed for its global Lipschitz continuity, positivity, existence and uniqueness of solutions. Second, the analysis is extended to a more complex and realistic model. Numerical simulations using Euler-Maruyama method are employed to demonstrate the long term dynamics. It was found that the noise added to the herbivore population resulted more change in the dynamics than the noise added to the plant population (food source). Ignoring the environmental noise could make the land management and wild life conservation not to maintain their goals.

Bifurcation and Stability Analysis of a System of Fractional-Order Differential Equations for a Plant–Herbivore Model with Allee Effect

This article concerns establishing a system of fractional-order differential equations (FDEs) to model a plant–herbivore interaction. Firstly, we show that the model has non-negative solutions, and then we study the existence and stability analysis of the constructed model. To investigate the case according to a low population density of the plant population, we incorporate the Allee function into the model. Considering the center manifold theorem and bifurcation theory, we show that the model shows flip bifurcation. Finally, the simulation results agree with the theoretical studies.