Stable coexistence of an invasive plant and biocontrol agent: a parameterized coupled plant-herbivore model

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 ς 1 , ϱ 1 , γ 1 and ϒ 1 , discrete-time plant-herbivore model has boundary and interior fixed points: c 1 = ( 0 , 0 ) , c 2 = ( ς 1 - 1 ϱ 1 , 0 ) and c 3 = ( ϒ 1 ( 1 - γ 1 ) 2 γ 1 - 1 , γ 1 ( 2 ς 1 + ϱ 1 ϒ 1 - 2 ) - ϱ 1 ϒ 1 + 1 - ς 1 2 γ 1 - 1 ) respectively. Then by linear stability theory, local dynamics with different topological classifications are investigated at fixed points: c 1 = ( 0 , 0 ) , c 2 = ( ς 1 - 1 ϱ 1 , 0 ) and c 3 = ( ϒ 1 ( 1 - γ 1 ) 2 γ 1 - 1 , γ 1 ( 2 ς 1 + ϱ 1 ϒ 1 - 2 ) - ϱ 1 ϒ 1 + 1 - ς 1 2 γ 1 - 1 ) . Our investigation uncovers that the boundary equilibrium c 2 = ( ς 1 - 1 ϱ 1 , 0 ) experiences a transcritical bifurcation, whereas the unique positive steady-state c 3 = ( ϒ 1 ( 1 - γ 1 ) 2 γ 1 - 1 , γ 1 ( 2 ς 1 + ϱ 1 ϒ 1 - 2 ) - ϱ 1 ϒ 1 + 1 - ς 1 2 γ 1 - 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.

Exploring dynamics of plant-herbivore interactions: bifurcation analysis and chaos control with Holling type-II functional response

In this study, we examine the plant-herbivore discrete model of apple twig borer and grape vine interaction, with a particular emphasis on the extended weak-predator response to Holling type-II response. We explore the dynamical and qualitative analysis of this model and investigate the conditions for stability and bifurcation. Our study demonstrates the presence of the Neimark-Sacker bifurcation at the interior equilibrium and the transcritical bifurcation at the trivial equilibrium, both of which have biological feasibility. To avoid unpredictable outcomes due to bifurcation, we employ chaos control methods. Furthermore, to support our theoretical and mathematical findings, we develop numerical simulation techniques with examples. In summary, our research enhances the comprehension of the dynamics pertaining to interactions between plants and herbivores in the context of discrete-time population models.

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.

Variation in reproductive mode across the latitudinal range of invasive Russian knapweed

For invading species, reproduction is a critical determinant of population establishment as well as spread into new areas. When species have multiple modes of reproduction, the prevalence of different modes can influence management decisions. We used genetic markers to determine the prevalent method of recruitment for invasive Russian knapweed (Acroptilon repens). This species forms patches and can spread by both rhizomic growth and seed from outcrossing. We found no shared genotypes between 41 western North American populations, indicating at the macroscale, Russian knapweed is spreading via seed to distant locations. We also examined drivers of reproductive mode by comparing clonality with large-scale environmental factors across the invasion. We found a correlation between latitude and clonal versus seed reproduction, with clonality higher in northern latitude populations. This trend was associated most parsimoniously with decreasing maximum annual temperature and 30-year average of available growing degree days, and increasing soil organic carbon content. These results have management implications: if not properly temporally implemented, grazing or herbicide applications that create open spaces for recruitment may increase the likelihood of Russian knapweed patch persistence through seed, and recently released galling biological control agents in North America may be less effective in northern latitudes where Russian knapweed spread by seed is less prevalent.

Altered precipitation and root herbivory affect the productivity and composition of a mesic grassland

BACKGROUND

Climate change models predict changes in the amount, frequency and seasonality of precipitation events, all of which have the potential to affect the structure and function of grassland ecosystems. While previous studies have examined plant or herbivore responses to these perturbations, few have examined their interactions; even fewer have included belowground herbivores. Given the ecological, economic and biodiversity value of grasslands, and their importance globally for carbon storage and agriculture, this is an important knowledge gap. To address this, we conducted a precipitation manipulation experiment in a former mesic pasture grassland comprising a mixture of C₄ grasses and C₃ grasses and forbs, in southeast Australia. Rainfall treatments included a control [ambient], reduced amount [50% ambient] and reduced frequency [ambient rainfall withheld for three weeks, then applied as a single deluge event] manipulations, to simulate predicted changes in both the size and frequency of future rainfall events. In addition, half of all experimental plots were inoculated with adult root herbivores (Scarabaeidae beetles).

RESULTS

We found strong seasonal dependence in plant community responses to both rainfall and root herbivore treatments. The largest effects were seen in the cool season with lower productivity, cover and diversity in rainfall-manipulated plots, while root herbivore inoculation increased the relative abundance of C₃, compared to C₄, plants.

CONCLUSIONS

This study highlights the importance of considering not only the seasonality of plant responses to altered rainfall, but also the important role of interactions between abiotic and biotic drivers of vegetation change when evaluating ecosystem-level responses to future shifts in climatic conditions.

Post-release monitoring in classical biological control of weeds: assessing impact and testing pre-release hypotheses

While various aspects of classical biological control (CBC) of weeds, including non-target risk assessment, have been continuously improved in the past few decades, post-release monitoring remains neglected and underfunded. Detailed assessments of the population, community and ecosystem outcomes of CBC introductions, including reasons for success/failure and absence or evidence of non-target effects are generally lacking or fragmentary. Here we review recent advances in understanding the demography of biological control agents released into a novel environment, their impact on the target weed and on non-target species, and the consequences for the resident plant and animal communities and ecosystem functioning, including the restoration of ecosystem services. We argue that post-release monitoring of CBC programs offers unique but largely underutilized opportunities to improve our understanding of CBC outcomes and to inform management and decision-makers on when and how CBC should be integrated with other management options to enhance ecosystem restoration.

Bifurcation analysis and chaos control for a plant-herbivore model with weak predator functional response

The interaction between plants and herbivores is one of the most fundamental processes in ecology. Discrete-time models are frequently used for describing the dynamics of plants and herbivores interaction with non-overlapping generations, such that a new generation replaces the old at regular time intervals. Keeping in view the interaction of the apple twig borer and the grape vine, the qualitative behaviour of a discrete-time plant-herbivore model is investigated with weak predator functional response. The topological classification of equilibria is investigated. It is proved that the boundary equilibrium undergoes transcritical bifurcation, whereas unique positive steady-state of discrete-time plant-herbivore model undergoes Neimark-Sacker bifurcation. Numerical simulation is provided to strengthen our theoretical discussion.

An Alternative Perspective for the Theory of Biological Control

Importation biological control represents the planned introduction of a specialist natural enemy from the region of origin of an invasive pest or weed. For this study, the author considered why attempts to develop a predictive theory for biological control have been misguided and what future directions might be more promising and effective. Despite considerable interest in the theory of consumer–resource population dynamics, such theory has contributed little to improvements in the success of biological control due to a focus on persistence and equilibrium dynamics rather than establishment and impact. A broader consideration of invasion biology in addition to population ecology offers new opportunities for a more inclusive theory of biological control that incorporates the demographic and genetic processes that more specifically address the establishment and impact of introduced natural enemies. The importance of propagule size and genetic variance for successful establishment, and of contributions to host population growth, relative population growth rates, interaction strength, and coevolution for suppression of host abundance are discussed as promising future directions for a theory of biological control.

Co-existence thresholds in the dynamics of the plant–herbivore interaction with Allee effect and harvest

In the interaction between plants and herbivores that live in the same ecosystem, understanding the conditions in which co-existence equilibrium occurs answers a major question in Ecology. In this interaction, plants serve as food for herbivores on the food chain. Then the livelihood of herbivores highly depends on the availability of food, in this case the availability of plants. Moreover, the abundance of the plant density alone does not guarantee the non-extinction of the herbivore population as they are assumed to reproduce sexually. With this motivation, in this paper a predator–prey mathematical model is reformulated such that the death rate of the herbivore population is dependent on the plant density and their emergence is also governed by the Allee effect. Using the mathematical theory of dynamical system, threshold conditions are obtained for the non-extinction of the herbivore population and a trapping region is obtained to ensure co-existence of the population. Moreover, it has been shown that the dynamics of the population is significantly sensitive to the feeding rate and the harvest rate of the herbivore population.

Native insect herbivory overwhelms context dependence to limit complex invasion dynamics of exotic weeds

Understanding the role of consumers in density-dependent plant population dynamics is a long-standing goal in ecology. However, the generality of herbivory effects across heterogeneous landscapes is poorly understood due to the pervasive influence of context-dependence. We tested effects of native insect herbivory on the population dynamics of an exotic thistle, Cirsium vulgare, in a field experiment replicated across eight sites in eastern Nebraska. Using hierarchical Bayesian analysis and density-dependent population models, we found potential for explosive low-density population growth (λ > 5) and complex density fluctuations under herbivore exclusion. However, herbivore access drove population decline (λ < 1), suppressing complex fluctuations. While plant-herbivore interaction outcomes are famously context-dependent, we demonstrated that herbivores suppress potentially invasive populations throughout our study region, and this qualitative outcome is insensitive to environmental context. Our novel use of Bayesian demographic modelling shows that native insect herbivores consistently prevent hard-to-predict fluctuations of weeds in environments otherwise susceptible to invasion.

Potential impacts of tolerance to herbivory on population dynamics of a monocarpic herb

PREMISE OF THE STUDY

Mammalian herbivores, particularly white-tailed deer, can have a major impact on plant abundance and distribution. However, plants can tolerate herbivory by increasing seed production or seed quality. We used the monocarpic perennial Prenanthes roanensis to examine tolerance to mammalian herbivory through seed quality and modeled the effects of tolerance on population growth rate.

METHODS

We examined seed quality (proportion of viable seeds, seed mass, germination, and seedling size) on damaged and undamaged plants to determine the extent to which plants tolerate herbivory. We then varied seed quality parameters over a range of values in population models to compare population growth rates under "no-tolerance" conditions (herbivory, but no tolerance) to those under "tolerance" conditions.

KEY RESULTS

In most populations, plants damaged by herbivores had a greater proportion of viable seeds per plant or a greater probability of seed germination. Incorporating observed tolerance into population models did not significantly increase population growth rate. However, at low germination rates, increased germination of seeds from damaged plants has the potential to significantly increase population growth rate.

CONCLUSIONS

Damaged plants can compensate for loss of reproductive heads by increasing seed viability and germination rates in the remaining seeds. This study is one of the first to demonstrate that tolerance through seed quality has the potential to affect population growth rate. Our results suggest that incorporating tolerance into population models may help elucidate mechanisms by which plant populations persist despite herbivory.

Release from belowground enemies and shifts in root traits as interrelated drivers of alien plant invasion success: a hypothesis

Our understanding of the interrelated mechanisms driving plant invasions, such as the interplay between enemy release and resource‐acquisition traits, is biased by an aboveground perspective. To address this bias, I hypothesize that plant release from belowground enemies (especially fungal pathogens) will give invasive plant species a fitness advantage in the alien range, via shifts in root traits (e.g., increased specific root length and branching intensity) that increase resource uptake and competitive ability compared to native species in the alien range, and compared to plants of the invader in its native range. Such root‐trait changes could be ecological or evolutionary in nature. I explain how shifts in root traits could occur as a consequence of enemy release and contribute to invasion success of alien plants, and how they could be interrelated with other potential belowground drivers of invasion success (allelopathy, mutualist enhancement). Finally, I outline the approaches that could be taken to test whether belowground enemy release results in increased competitive ability and nutrient uptake by invasive alien plants, via changes in root traits in the alien range.

A seasonal, density-dependent model for the management of an invasive weed

The population effects of harvest depend on complex interactions between density dependence, seasonality, stage structure, and management timing. Here we present a periodic nonlinear matrix population model that incorporates seasonal density dependence with stage-selective and seasonally selective harvest. To this model, we apply newly developed perturbation analyses to determine how population densities respond to changes in harvest and demographic parameters. We use the model to examine the effects of popular control strategies and demographic perturbations on the invasive weed garlic mustard (Alliaria petiolata). We find that seasonality is a major factor in harvest outcomes, because population dynamics may depend significantly on both the season of management and the season of observation. Strategies that reduce densities in one season can drive increases in another, with strategies giving positive sensitivities of density in the target seasons leading to compensatory effects that invasive species managers should avoid. Conversely, demographic parameters to which density is very elastic (e.g., seeding survival, second-year rosette spring survival, and the flowering to fruiting adult transition for maximum summer densities) may indicate promising management targets.

Estimating development rate and thermal requirements of Bactericera cockerelli (Hemiptera: Triozidae) reared on potato and tomato by using linear and nonlinear models

The temperature-dependent development of tomato psyllid (also called potato psyllid), Bactericera cockerelli (Sulc), was studied in the laboratory at seven constant temperatures (8, 10, 15, 20, 23, 27, and 31°C), 50-60% RH, and a photoperiod of 16:8 (L:D) h on leaves of whole potato (Solanum tuberosum L.) and tomato (Solanum lycopersicum L.) plants. Developmental time in days for immature stages and total development (egg to adult) on both host species were inversely proportional to temperature between 8 and 27°C but increased at 31°C. One linear and two nonlinear models were fitted to the data. The lower developmental thresholds, calculated using the linear model for egg, total nymph, and total development (from oviposition to adult emergence) were 7.9, 4.2, and 7.1°C (reared on potato) and 7.2, 5.3, and 7.5°C (reared on tomato), respectively. The thermal constant (K) for total development was 358 (reared on potato), and 368 (reared on tomato) degree-days (DD). Two nonlinear models, Briere and Lactin, fit the data well as measured by goodness-of-fit criteria, the residual sum of square (RSS) and Akaike information criterion (AIC). Temperature threshold parameters for these nonlinear models (T(0), T(opt), and T(max)) were estimated for eggs, total nymphal stages, and total development time (egg to adult). The Briere model is highly recommended for the description of temperature-dependent development of tomato psyllid. Results from this study will provide basic information on the biology of tomato psyllid and have potential for the development of predictive models of the seasonal progress of this invasive pest.

Variable effects of a generalist parasitoid on a biocontrol seed predator and its target weed

Biological control (the importation of enemies from an invader's native range) is often considered our best chance of controlling the most widespread invaders. Ideally, the agent reduces invader abundance to some acceptably low level, and the two coexist at low density with the agent providing continuous control over the long-term. But the outcome may be complicated when the agent is attacked by native predators and parasites. We used a spatially explicit, discrete-time, individual-based, coupled plant-seed predator-parasitoid model to estimate the impact of the biocontrol agent Eustenopus villosus (a seed predator) on the invasive, annual weed Centaurea solstitialis, both with and without the generalist parasitoid Pyemotes tritici. We estimated the agent's ability to reduce plant density, spread rate, and population growth rate over 50 years. We used long-term demographic data from two sites in central California, USA, to parameterize the model and assess how populations in different climatic zones might respond differently to the agent and the parasitoid. We found that the biocontrol agent reduced plant density (relative to predictions for an uncontrolled invasion), but its impact on the invader's spread rate was modest and inconsistent. The agent had no long-term impact on population growth rate (lambda). Parasitism caused a trophic cascade, the strength of which varied between sites. At our coastal site, the parasitoid entirely eliminated the impact of the agent on the plant. At our Central Valley site, even when parasitized, the agent significantly reduced plant density and spread rate over several decades (although to a lesser degree than when it was not parasitized), but not invader lambda. Surprisingly, we also found that the length of time the invader was allowed to spread across the landscape prior to introducing the agent (5, 25, or 50 years) had little influence over its ability to control the weed in the long-term. This is encouraging news for land managers attempting to control invasive plants that have already established widespread, high-density populations. Unfortunately, our results also show that attack by the native generalist parasitoid had a larger influence over how effectively the agent reduced invader performance.

Food limitation and insect outbreaks: complex dynamics in plant?herbivore models

1. The population dynamics of many herbivorous insects are characterized by rapid outbreaks, during which the insects severely defoliate their host plants. These outbreaks are separated by periods of low insect density and little defoliation. In many cases, the underlying cause of these outbreaks is unknown. 2. Mechanistic models are an important tool for understanding population outbreaks, but existing consumer-resource models predict that severe defoliation should happen much more often than is seen in nature. 3. We develop new models to describe the population dynamics of plants and insect herbivores. Our models show that outbreaking insects may be resource-limited without inflicting unrealistic levels of defoliation. 4. We tested our models against two different types of field data. The models successfully predict many major features of natural outbreaks. Our results demonstrate that insect outbreaks can be explained by a combination of food limitation in the herbivore and defoliation and intraspecific competition in the host plant.

Demographic models inform selection of biocontrol agents for garlic mustard (Alliaria petiolata)

Nonindigenous invasive plants pose a major threat to natural communities worldwide. Biological control of weeds via selected introduction of their natural enemies can affect control over large spatial areas but also risk nontarget effects. To maximize effectiveness while minimizing risk, weed biocontrol programs should introduce the minimum number of host-specific natural enemies necessary to control an invasive nonindigenous plant. We used elasticity analysis of a matrix model to help inform biocontrol agent selection for garlic mustard (Alliaria petiolata (M. Bieb.) Cavara and Grande). The Eurasian biennial A. petiolata is considered one of the most problematic invaders of temperate forests in North America. Four weevil species in the genus Ceutorhynchus (Coleoptera: Curculionidae) are currently considered potential biocontrol agents. These species attack rosettes (C. scrobicollis), stems (C. roberti, C. alliariae), and seeds (C. constrictus) of A. petiolata. Elasticity analyses using A. petiolata demographic parameters from North America indicated that changes in the rosette-to-flowering-plant transition and changes in fecundity consistently had the greatest impact on population growth rate. These results suggest that attack by the rosette-feeder C. scrobicollis, which reduces overwintering survival, and seed or stem feeders that reduce seed output should be particularly effective. Model outcomes differed greatly as A. petiolata demographic parameters were varied within ranges observed in North America, indicating that successful control of A. petiolata populations may occur under some, but not all, conditions. Using these a priori analyses we predict: (1) rosette mortality and reduction of seed output will be the most important factors determining A. petiolata demography; (2) the root-crown feeder C. scrobicollis will have the most significant impact on A. petiolata demography; (3) releases of single control agents are unlikely to control A. petiolata across its full range of demographic variability; (4) combinations of agents that simultaneously reduce rosette survival and seed production will be required to suppress the most vigorous A. petiolata populations. These predictions can be tested using established long-term monitoring sites coupled with a designed release program. If demographic models can successfully predict biocontrol agent impact on invasive plant populations, a continued dialogue and collaboration between empirical and theoretical approaches may be the key to the development of successful biocontrol tactics for plant invaders in the future.

DETECTION OF DENSITY DEPENDENCE REQUIRES DENSITY MANIPULATIONS AND CALCULATION OF λ

To investigate density-dependent population regulation in the perennial bunchgrass Bouteloua rigidiseta, we experimentally manipulated density by removing adults or adding seeds to replicate quadrats in a natural population for three annual intervals. We monitored the adjacent control quadrats for 14 annual intervals. We constructed a population projection matrix for each quadrat in each interval, calculated lambda, and did a life table response experiment (LTRE) analysis. We tested the effects of density upon lambda by comparing experimental and control quadrats, and by an analysis of the 15-year observational data set. As measured by effects on lambda and on N(t+1/Nt in the experimental treatments, negative density dependence was strong: the population was being effectively regulated. The relative contributions of different matrix elements to treatment effect on lambda differed among years and treatments; overall the pattern was one of small contributions by many different life cycle stages. In contrast, density dependence could not be detected using only the observational (control quadrats) data, even though this data set covered a much longer time span. Nor did experimental effects on separate matrix elements reach statistical significance. These results suggest that ecologists may fail to detect density dependence when it is present if they have only descriptive, not experimental, data, do not have data for the entire life cycle, or analyze life cycle components separately.