Predator Populations Research Articles

Dynamical behaviors of autonomous and nonautonomous models of generalist predator–prey system with fear, mutual interference and nonlinear harvesting

In this study, we explore a predator–prey model to evaluate the impacts of fear, mutual interference, and harvesting, as these factors play crucial roles in modifying population dynamics, trophic cascades, and species interactions. Additionally, we enhance the model to incorporate the seasonality of specific ecological factors across time. The autonomous and associated nonautonomous models have undergone thorough mathematical analysis and extensive numerical simulations in the course of the study. We conduct a fundamental differential sensitivity analysis to discern the key parameters influencing both prey and predator populations within the ecosystem. Through numerical investigations of the autonomous model, we notice that in the presence (absence) of harvesting, the mutual interference among predators destabilizes (stabilizes) the coexisting equilibrium. The fear effect significantly reduces the densities of prey and predator populations, and destabilizes the coexisting equilibrium by producing multiple stability switches. The survival of predators substantially depends on the focal prey if their growth from other food sources is relatively low. Notably, the harvesting of predators is beneficial and promotes the coexistence of species only when their growth from other food sources is too much. Moreover, the system shows bistability, tristability, and multiple limit cycles under identical ecological conditions. We also show various phenomena that manifest within the seasonally forced model, including the emergence of periodic solutions, higher periodic solutions, bursting patterns, and chaotic dynamics. Remarkably, the influence of seasonal forcing possesses the capability to govern the chaotic dynamics by giving rise to an exceedingly quasi-periodic arrangement in the prey and predator densities.

Impact of wind flow and global warming in the dynamics of prey–predator model

Prey–predator interaction is among the most crucial aspects of ecology. In this paper, the dynamics of a prey–predator system in which the prey and predator populations experience a birth rate reduction due to global warming and the wind flow effect on them are investigated. The system’s well-posedness before presenting findings about the existence of potential equilibrium points is studied. Conditions under which a different number of interior equilibria exist are also derived here. It is very important to know that when the positive equilibrium point is stable, this reflects positively on the ecosystem, its continuity and persistence. The conditions that have been identified are conditions that preserve the ecosystem from extinction when environmental influences such as global warming and wind speed are present. In addition, the local stability study around all equilibrium points and established the transversality criteria for the occurrence of transcritical bifurcations and Hopf bifurcations is conducted. The existence of different equilibrium points numerically by altering the prey’s global warming effect is illustrated , also showcasing the model’s dynamic character from a mathematical perspective. For significant amounts of global warming, its notice that the system exhibits periodic behavior. Thus, ecologists can approximate the parameters necessary to investigate and obtain such crucial findings regarding nonlinear systems with the aid of the current study.

Complex dynamics of a discrete-time Leslie-Gower predator-prey system with herd behavior and slow-fast effect on predator population

Complex dynamics of a discrete-time Leslie-Gower predator-prey system with herd behavior and slow-fast effect on predator population

Carotenoid skin ornaments as flexible indicators of male foraging behavior in a marine predator: Variation among Mexican colonies of brown booby (Sula leucogaster)

Carotenoid-dependent ornaments can reflect animals’ diet and foraging behaviors. However, this association should be spatially flexible and variable among populations to account for geographic variation in optimal foraging behaviors. We tested this hypothesis using populations of a marine predator (the brown booby, Sula leucogaster) that forage across a gradient in ocean depth in and near the Gulf of California. Specifically, we quantified green chroma for two skin traits (foot and gular color) and their relationship to foraging location and diet of males, as measured via global positioning system tracking and stable carbon isotope analysis of blood plasma. Our three focal colonies varied in which foraging attributes were linked to carotenoid-rich ornaments. For gular skin, our data showed a shift from a benthic prey-green skin association in the shallow waters in the north to a pelagic prey-green skin association in the deepest waters to the south. Mean foraging trip duration and distance of foraging site from coast also predicted skin coloration in some colonies. Finally, brown booby colonies varied in which trait (foot versus gular skin color) was associated with foraging metrics. Overall, our results indicate that male ornaments reflect quality of diet and foraging–information that may help females select mates who are adapted to local foraging conditions and therefore, are likely to provide better parental care. More broadly, our results stress that diet-dependent ornaments are closely linked to animals’ environments and that we cannot assume ornaments or ornament signal content are ubiquitous within species, even when ornaments appear similar among populations.

Does mutual interference stabilize prey–predator model with Bazykin–Crowley–Martin trophic function?

We investigated a system of ordinary differential equations that describes the dynamics of prey and predator populations, taking into account the Allee effect affecting the reproduction of the predator population, and mutual interference amongst predators, which is modeled with the Bazykin–Crowley–Martin (BCM) trophic function. Bifurcation analysis revealed a rich spectrum of bifurcations occurring in the system. In particular, analytical conditions for the saddle–node, Hopf, cusp, and Bogdanov–Takens bifurcations were derived for the model parameters, quantifying the strength of the predator interference, the Allee effect, and the predation efficiency. Numerical simulations verify and illustrate the analytical findings. The main purpose of the study was to test whether the mutual interference in the model with BCM trophic function provides a stabilizing or destabilizing effect on the system dynamics. The obtained results suggest that the model demonstrates qualitatively the same pattern concerning varying the interference strength as other predator-dependent models: both low and very high interference levels increase the risk of predator extinction, while moderate interference has a favorable effect on the stability and resilience of the prey–predator system.

Precipitation seasonality determines the potential distribution of Hyaena hyaena in Saudi Arabia: Towards conservation planning

Hyaena hyaena is a carnivorous mammal that inhabits a broad variety of habitat types and ecological niches, H. hyaena spread in the Arabian Peninsula excluding Rub Al Khali and the desert of Nafud. In Saudi Arabia, it can be found commonly in the western mountains and the open land near agricultural areas such as valleys and lava fields (Harat). However, the H. hyaena is considered to be globally “Near Threatened” (NC) according to (IUCN) Red List as a consequence of a drop in carrion as a result of a reduction in the population of big predators and their prey, eradication practices, and fragmentation and loss of their habitat. The current study seeks to evaluate the distribution of H. hyaena in Saudi Arabia in relation to a number of environmental parameters, as well as assess the possible appropriateness of different habitats for it under climate change scenarios, and provide significant insights for future conservation strategies. The species distribution models (SDMs) tools were employed to map and monitor the distribution of H. hyaena, and then provide a prediction of species distribution patterns. Multicollinearity analysis of the total 19 bioclimatic variables and topographic roughness index resulted in seven uncorrelated variables with VIF < 5, which had been used in the ensemble modeling. The result of our study showed that the decrease in seasonal precipitation will lead to a decrease in the probability of the presence of H. hyaena in Saudi Arabia and The probability of finding H. hyaena in Saudi Arabia decreases as the driest quarter and wettest quarter mean temperature decreases.

Model Predator-Prey Leslie-Gower dengan Fungsi Respon Sokol-Howell dan Perilaku Anti Predator

This study discusses the Leslie-Gower predator-prey model with the Sokol-Howell response function and anti-predator behavior. It is assumed that prey has anti-predator behavior that aims to reduce the risk of predation and not as an attempt by prey to find food. This study aims to formulate a Leslie-Gower predator-prey model with the Sokol-Howell response function and anti-predator behavior, analyze the model's equilibrium point and model interpretation. Stability analysis was carried out using the linearization method. The type of stability is determined based on the characteristic eigenvalues ​​obtained using Routh-Hurwitz criteria. The results of the analysis of the equilibrium point show that prey populations will exist and predators will become extinct if the anti-predator coefficient is greater than the intrinsic growth coefficient of predators, while prey and predator populations will always exist if the intrinsic growth coefficient of predators is greater than the anti-predator coefficient and fulfills other conditions required. Based on the numerical simulations performed, the interpretation is that an enlarged anti-predator coefficient increases the number of prey populations until they approach the carrying capacity, while predator populations decrease significantly and over time experience extinction.

Prey-mediated effects of Mpp51Aa2-producing cotton on longevity and reproduction of Orius majusculus

Genetically engineered (GE) cotton event MON 88702, producing Mpp51Aa2 (previously mCry51Aa2) from Bacillus thuringiensis (Bt), controls sucking pests, such as Lygus spp. (Hemiptera: Miridae) and thrips (Thysanoptera). Ingesting high doses of the insecticidal protein resulted in adverse effects on life table parameters of beneficial, predatory Orius spp. (Hemiptera: Anthocoridae). This triggered laboratory studies with more realistic food treatments, including different combinations of prey types with and without Bt protein to further characterize risks to this important group of non-target organisms. In this work, exclusive feeding of frozen spider mites (Tetranychus urticae, Acari: Tetranychidae) from Bt cotton confirmed adverse effects on longevity and fecundity of O. majusculus adults. Alternate feeding of Bt protein-containing spider mites and Bt-free Ephestia kuehniella (Lepidoptera: Pyralidae) eggs mitigated effects on longevity, but not on fecundity. When living larvae of Spodoptera littoralis (Lepidoptera: Noctuidae) from Bt cotton were fed to the predators, however, no effects on longevity and reproduction of female O. majusculus were observed, despite the fact that Bt protein concentrations in larvae were almost as high as concentrations in spider mites. When a diverse mix of prey species with various Bt protein concentrations is consumed in the field, it is unlikely that exposure of Orius spp. to Mpp51Aa2 is high enough to exert adverse effects on predator populations. MON 88702 cotton may thus be a valuable tool for integrated management of sucking pests.

Livestock Depredation by Large Carnivores and Human-Wildlife Conflict in Two Districts of Balochistan Province, Pakistan.

Livestock herding is a vital practice in Balochistan, contributing to the economy and culture. The livestock sector is significant in Balochistan, providing 20% of the national stock. Large predators and their prey species, including livestock, have coexisted in these mountainous landscapes for centuries. The aim of the present research is to investigate the impacts of livestock depredation by large predators on livelihoods and predator conservation in two districts of Balochistan, Pakistan. A human-carnivore conflict survey was conducted from July to September 2019, collecting data from 311 residents in a selected study area. Large predators in the study area preyed on a total of 876 livestock during a one-year period, including 560 goats, 292 sheep, 19 cows, and 5 donkeys. The gray wolf is the leading predator, responsible for 66.3% of livestock depredation, followed by the caracal (24.3%), Asiatic jackal (8.9%), and striped hyena (0.6%). The total economic loss was USD 78,694. Overall, 80% of respondents had a negative perception of wolves compared to 24.4% for caracals. Only 20.6% of respondents knew about the importance of conserving carnivores. Livestock depredation by carnivores in the study area created a negative perception of these animals among people. There is a lack of awareness about the importance of conserving carnivore species and their role in the ecosystem. This lack of understanding has ultimately led to detrimental effects on predator populations. It is imperative to raise awareness among people about the ecological significance of carnivores through community meetings, seminars in educational institutions, and providing basic education to herders about effective livestock guarding practices.

Response of diademed sifaka (Propithecus diadema) to fosa (Cryptoprocta ferox) predation in the Betampona Strict Nature Reserve, Madagascar.

Large-bodied mammals living in fragmented habitats are at higher risk of extinction, and such risk can be influenced by ecological factors such as predator-prey system dynamics. These dynamics can be particularly complex for conservation management when one endangered species preys on another endangered species in an isolated or poor-quality habitat. Here we describe predation events observed over 19 months that involved two threatened species: the largest carnivore in Madagascar, the fosa (Cryptoprocta ferox), and three groups of diademed sifaka (Propithecus diadema) in the Betampona Strict Nature Reserve. This site is a 22 km2 low-altitude rainforest that is surrounded by agricultural land and isolated from larger forest corridors. We aim to (1) assess the behavioral changes of P. diadema in response to fosa attacks and identify any antipredator strategies that they adopted, and (2) quantify the frequency of fosa attacks and the predation impact on the sifaka population. We report five direct observations of fosa predation attempts (one successful), the discovery of a dead sifaka with evidence of fosa predation, and the disappearance of three individuals. We describe the observed attacks and compare the sifaka activity budgets and movement patterns before and after the events. To escape the predator, sifakas fled short distances, hid, and remained vigilant. The impact of predation, combined with low reproductive rates and potentially high inbreeding of this isolated diademed sifaka population, could affect the survival of this species in Betampona.Given the compounding effects of habitat isolation and high hunting pressure, community-specific conservation strategies should incorporate predator-prey dynamics via longitudinal monitoring of predator and prey population densities and quantifying the predation pressure between them.

Spatiotemporal and Trade-Off Dynamics in Prey–Predator Model with Domed Functional Response and Fear Effect

In the ecological scenario, predators often risk their lives pursuing dangerous prey, potentially reducing their chances of survival due to injuries. Prey, on the other hand, try to strike a balance between reproduction rates and safety. In our study, we introduce a two-dimensional prey–predator model inspired by Tostowaryk’s work, specifically focusing on the domed-shaped functional response observed in interactions between pentatomid predators and neo-diprionid sawfly larvae. To account for the varying effectiveness of larval group defense, we incorporate a new component into the response equation. Our investigation delves into predator trade-off dynamics by adjusting the predator’s mortality rate to reflect losses incurred during encounters with dangerous prey and prey’s trade-off between safety and reproduction rate incorporating this domed-shaped functional response. Our model demonstrates bistability and undergoes various bifurcations, including transcritical, saddle-node, Hopf, Bogdanov–Takens, and Homoclinic bifurcations. Critical parameters impact both predator and prey populations, potentially leading to predator extinction if losses due to dangerous prey encounters become excessive, highlighting the risks predators face for their survival. Furthermore, the efficacy of group defense mechanisms can further endanger predators. Expanding our analysis to a spatially extended model under different perturbations, we explore Turing instability to explain the relationship between diffusion and encounter parameters through both stationary and dynamic pattern formation. Sensitivity to initial conditions uncovers spatiotemporal chaos. These findings provide valuable insights into comprehending the intricate dynamics of prey–predator interactions within ecological systems.

Effectively Incorporating Small Reserves into National Systems of Protected and Conserved Areas

Received wisdom has argued that large protected areas are superior to small reserves, based on island biogeography theory, economies of scale, and the need to sustain viable populations of top predators and other large ranging or dispersive species. But this position overlooks evidence that, for many species, strategically placed smaller reserves are extremely important, especially in areas highly altered by humans. Many countries are reluctant or unable to designate additional large protected areas. We provide evidence that carefully designed support networks of smaller protected areas can be an important complement to activities to reach the Global Biodiversity Framework’s target of 30% of the planet in protected and conserved areas by 2030. We identify seven benefits from small reserves, when correctly located and well-managed: (1) conserving critical habitat of range-limited or relic species; (2) conserving remaining areas of sensitive or threatened habitat in altered ecosystems; (3) conserving habitat for sensitive, time-limited lifecycle stages, such as raptor nesting sites and fish spawning grounds; (4) maintaining and enabling connectivity by providing stepping stones of suitable habitat through inhospitable ecosystems; (5) providing increased protection for critical habitat within Category V protected landscapes and seascapes to boost their overall conservation potential; (6) taking advantage of conservation opportunities at cultural sites, sacred natural sites, and other faith-based sites in transformed landscapes; (7) integrating different management approaches and governance types in a range of connected small reserves to multiply conservation impacts. We propose a typology based on these benefits that can guide steps for policy makers to help plan and monitor small reserves in area-based conservation efforts. Using these principles, the role of small reserves in area-based conservation efforts can be further enhanced.

Dynamical of Prey Refuge in a Diased Predator-Prey Model with Intraspecific Competition for Predator

The predator-prey model described is a population growth model of an eco-epidemiological system with prey protection and predator intraspecific traits. Predation interactions in predator species use response functions. The aim of this research is to examine the local stable balance point and look at the characteristics of species resulting from mathematical modeling interventions. Review of balance point analysis, numerical simulation and analysis of given trajectories. The research results show the shape of the model which is arranged with a composition of 5 balance points. There is one rational balance point to be explained, using the Routh-Hurwitz criterion, . The characteristic equation and associated eigenvalues in the mathematical model are the local asymptotically stable balance points. In the trajectory analysis, local stability is also shown by the model formed. There are differences for each population to reach its point of stability. The role of prey protection behavior is very effective in suppressing the spread of disease. Meanwhile, intraspecific predator interactions are able to balance the decreasing growth of prey populations. If we increase the intraspecific interaction coefficient, we can be sure that the growth of the prey population will both increase significantly. When the number of prey populations increases significantly, of course disease transmission and prey protection become determining factors, the continuation of the model in exosite interactions. In prey populations and susceptible prey to infection, growth does not require a long time compared to the growth of predator populations. The time required to achieve stable growth is rapid for the prey species. Although prey species' growth is more fluctuating compared to predator populations. Predatory species are more likely to be stable from the start of their growth. The significance of predatory growth is only at the beginning of growth, while after that it increases slowly and reaches an ideal equilibrium point. Each species has its own characteristics, so extensive studies are needed on more complex forms of response functions in further research.

Performance evaluation of a prototype for the defense against wolf attacks on livestock animals

This study seeks to evaluate a prototype's effectiveness in safeguarding livestock against wolf attacks. With an increasing imperative to protect livestock from predation, the prototype's performance was systematically examined under diverse conditions. The study primarily aimed at assessing the prototype's ability to detect wolf attacks by analyzing noise variations inherent to predator assaults. Simultaneously, the prototype aimed to mitigate livestock casualties and foster coexistence between wolves and livestock. A series of controlled experiments were meticulously carried out, replicating real-world wolf encounter scenarios. The findings yield valuable insights into the prototype's practical utility and its potential to mitigate conflicts between predator populations and livestock farming. Overall, this research contributes to advancing innovative strategies for sustainable cohabitation between wildlife and agriculture.

On the slow–fast dynamics of a tri‐trophic food chain model with fear and Allee effects

A three‐species food chain model with Allee and fear effects on both prey and predator is investigated in this work. The model consists of ODEs describing the evolution and interaction of prey, predator, and top predator populations in a biological system. We assume that the intrinsic growth rate of the prey is much smaller than a threshold value determined by other biological parameters of the model. Consequently, the model is transformed into a singularly perturbed system that acts on two different time scales, ranging from slow for both prey and predator to fast for the top predator. Positivity and boundedness of the solutions of the model along with the stability for its equilibria are investigated. The model possesses the exact invariant manifold with changing stability (black swan) and exhibits the canard phenomenon. Some numerical simulations are performed to validate our analysis.

Nanofertilizers and plant metabolites: Investigating their influence on trophic interactions in the cucumber-melon aphid-green lacewing

The role of fertilizers in increasing nutritional content and making host plants more attractive to herbivores has been repeatedly proved. This study investigated the effect of applying three levels of nano-chelated fertilizer (vegetables and summer crops specific fertilizer) (2, 3, and 4 kg/ha) on the levels of malondialdehyde (MDA), and phenylalanine ammonia-lyase (PAL), peroxidase (POD), polyphenol oxidase (PPO), catalase (CAT), ascorbic acid (ASA), flavonoid, and phenolic compounds in cucumber plant (Cucumis sativus L.). Additionally, the research surveyed the influence of fertilization and cucumber metabolite compounds on the life history of Aphis gossypii (Glover) (Hemiptera: Aphididae) and Chrysoperla carnea (Stephens) (Neuroptera: Chrysopidae) in a tri-trophic context. Host plant content analysis showed that biotic stress resulting from the aphid infestation in cucumber plants increased the POD, PPO, CAT activity, and MDA contents. However, it decreased PAL activity, flavonoid, and phenol contents in the control treatment and 2, 3, and 4 kg/ha of nano-chelated fertilizer levels. Moreover, plant fertilization increased the herbivore population growth rates and decreased the predator population growth rates by affecting the nutritional contents and quality of the host plant. Accordingly, the aphid's intrinsic rate of increase (r) was ranged from 0.319 to 0.365 day−1. While finite rate of increase (λ) was estimated as 1.376 to 1.441 day−1 on control to 2 kg/ha nano-chelated fertilizer treatment. In addition, the intrinsic rate of increase of C. carnea was 0.116 to 0.131 and the finite rate of increase was 1.123 to 1.139 day−1 on 2 kg/ha nano-chelated fertilizer and control treatments, respectively. The maximum and minimum net reproductive rate (R0) values of the aphid were recorded on 3 kg/ha nano-chelated fertilizer and control treatments as 46.34 and 37.18 offsprings, respectively, whilst these values ranged from 145.51 on 2 kg/ha to 196.93 offsprings on 4 kg/ha regimes for the green lacewing. Notably, the utilization of 2, 3, and 4 kg/ha of nano-chelated fertilizer promotes cucumber defensive metabolites compounds such as PAL, POD, CAT, flavonoid, and ASA and induces the plant's resistance to melon aphids directly. It can be concluded that the utilization of nano-chelated fertilizer potentially can enhance cucumber plant resistance to A. gossypii by improving the antioxidant metabolite compounds, and be an effective part of integrated pest management programs.

Bifurcation analysis of a discrete Leslie–Gower predator–prey model with slow–fast effect on predator

Understanding and accounting for the slow–fast effect are crucial for accurately modeling and predicting the dynamics of predator–prey models, emphasizing the importance of considering the relative speeds of interacting populations in ecological research. This paper examines a predator–prey interaction to study its complex dynamics due to its slow–fast effect on predator populations. The occurrence and stability of equilibria are analyzed. The stability of positive fixed point is dependent on the slow–fast effect parameter , which must fall within a specific range when the generation gap is larger. The positive fixed point becomes unstable for bigger values of because the growth of predators is faster, resulting in the extinction of all prey. Smaller values of cause the positive fixed point to become unstable since the prey grows more quickly while the predator grows more slowly, ultimately causing the extinction of the predator. Moreover, it is shown that Leslie–Gower model experiences Neimark–Sacker and period‐doubling bifurcations at positive equilibrium point. In order to control bifurcation, hybrid control and feedback control methods are employed. Finally, analytical results are confirmed by numerical examples.

A Complex Dynamic of an Eco-Epidemiological Mathematical Model with Migration

In this paper, we propose an eco-epidemiological mathematical model in order to describe the effect of migration on the dynamics of a prey–predator population. The functional response of the predator is governed by the Holling type II function. First, from the perspective of mathematical results, we develop results concerning the existence, uniqueness, positivity, boundedness, and dissipativity of solutions. Besides, many thresholds have been computed and used to investigate the local and global stability results by using the Routh–Hurwitz criterion and Lyapunov principle, respectively. We have also established the appearance of limit cycles resulting from the Hopf bifurcation. Numerical simulations are performed to explore the effect of migration on the dynamic of prey and predator populations.

Geometrical and Numerical Analysis of Predator–Prey System Based on the Allee Effect in Predator

This study explores the complex dynamics of the predator–prey interactions, with a specific emphasis on the influence of the Allee effect on the predator population. We examined the fundamental mathematical characteristics of the model under consideration, such as the positivity of the system and the boundedness of the solutions. We investigated the equilibrium points and analyzed their stability using the Jacobi and Lyapunov methods. A comprehensive examination was carried out on the geometric properties of the dynamical system to compute the five invariants of the KCC theory. In particular, the deviation curvature tensor and its eigenvalues are investigated to demonstrate the behavior of the system stability. We have also obtained the necessary and sufficient conditions for the given set of parameters of the system in order to have the Jacobi stability (instability) near the equilibrium point. To visualize the dynamical behavior of the predator–prey model with the Allee effect in the predator density, numerical simulations were conducted. The investigation encompasses an examination of the system’s behavior from both geometric and numerical standpoints, with the objective of attaining a thorough comprehension using few examples.

Role of Allee and Fear for Controlling Chaos in a Predator–Prey System with Circulation of Disease in Predator

This paper explores a predator–prey system featuring fear and disease within the predator population,utilizing the Rosenzweig–MacArthur model with Holling type-II functional response. The primary focus lies in investigating the impact of a fear factor, wherein the prey’s growth rate is hindered due to predator-induced fear. Additionally, the model accounts for the spread of disease among predators,leading to a division between susceptible and infected predator subpopulations. The inclusion of an Allee effect in the susceptible predator further enriches the model. The study involves a thorough examination, encompassing local and global stability analysis as well as Hopf bifurcation analysis around the interior equilibrium point. Numerical simulations underscore a noteworthy observation: an escalation in interaction force propels the system into chaotic dynamics,marked by stable focus, limit cycles and period-doubling phenomena. A noteworthy finding pertains to the influence of the Allee parameter ([Formula: see text]) on chaotic dynamics. As the Allee parameter values increase, the system tends to stable focus through a sequence of chaotic states, period-doubling and limit cycles. Subsequently, the paper introduces the role of another pivotal parameter, the fear factor, into the chaotic dynamics. Intriguingly, chaos transforms into stable focus through diverse nonlinear phenomena, including period-doubling and limit cycles. This nuanced exploration of parameters sheds light on the intricate dynamics governing the predator–prey system, offering a comprehensive understanding of the interplay between fear, disease and ecological factors. So our observation throughout this paper that how chaos behaves here after one by one injection of our new features: fear factor and Allee parameter?