A predator-prey model with satiation and intraspecific competition

We present a new predator-prey model where, except for the prey growth, assumed to be logistic, we endeavor to give some behavioral justification to all elements of the predator-prey interaction. The functional response takes account of predator satiation and predator competition. It is supported by some experimental evidence. We distinguish two contributions to the numerical response: the positive part, proportional to the functional response, is the birth rate of predators; the negative part is the death rate due to hunger.Two outcomes are possible. If the prey are unable to grow fast enough to replace the amount killed by the predators, both species become extinct. In the opposite case, both populations stabilize at a constant population. At this equilibrium level, the prey are not abundant enough to satiate the predators.The predation rate that allows the highest predator population is one half of the ideal prey growth rate. A higher exploitation rate can allow higher populations only temporarily. Evolved predator behavior, reguges for the prey, or other mechanisms can explain this regulation.Two more population behaviors (cycles and predator extinction) can be obtained with a time-lag in one of the responses. This is shown in a separate paper.The model is structurally stable. It can thus withstand small environmental perturbations.     

A dynamic programming implemented resource competition game theoretic model

Resource competition is commonly occurred in animal populations and studied intensively by researchers. Previous studies have applied game theoretic model by finding Nash equilibrium to investigate this phenomenon. However computation of the Nash equilibrium requires an understanding of the payoff matrix that allocates the rewards received by players when they adopt each of the strategies in the game. In our study we present a dynamic programming implemented framework to compute 2 × 2 intraspecific finite resource allocation game's payoff matrix explicitly. We assume that two distinct types of individuals, aggressive and non-aggressive, are in the population. Then we divide the entire animal development period into three different stages: initialization, quasilinear growth and termination. Each stage for each type of players is specified with their own development coefficient, which determines how resource consumption could convert into strength as reward. Each player has equal and finite resource at the beginning of their development and fights against other players in the population to maximize its own potential reward. Based on these assumptions it is reasonable to use backward induction dynamic programming to compute payoff matrix. We present numerical examples for three different types of aggressive individuals and compute the payoff matrices correspondingly. Then we use the derived payoff matrices to determine the Nash equilibrium and Evolutionary Stable Strategy. Our research provide a framework for future quantitative studies on animal resource competition problems and could be expanded to n-players interspecific stochastic asymmetric resource allocation problem by changing some settings of dynamic programming formulation.     

A general model for tagging on multiple component fisheries: an integration of age-dependent reporting rates and mortality estimation

A major objective of analyzing multiple year tag return data in fisheries is to estimate fishing and natural mortality rates which may vary by age class and calendar year. To do this one needs to be able to estimate the reporting rates for the tags recovered. Some fisheries such as that for Southern Bluefin Tuna (Thunnus maccoyii) have multiple components with potentially different reporting rates for the tag returns. In this paper we develop a general model for multi-cohort, multi-year tag return analyses where there are multiple components to the fishery with potentially different reporting rates. We require the assumption that one component has a reporting rate of 100% (i.e., this could be the component of a boat based fishery where scientific observers are present). We show further how it is possible to partition the overall likelihood developed into two conditionally independent components. The first component of the likelihood is the standard multinomial likelihood that allows estimation of fishing and natural mortality rates. It uses the tag return matrix summed over all the components of the fishery. It requires an average reporting rate for the tag returns (where the average reporting rate is a weighted average of the individual reporting rates of the different components). The second component is also multinomial for the individual component tag returns and allows us to estimate individual component reporting rates. However, this requires that we augment our second component tag return likelihood with a catch data likelihood for the corresponding components. The methodology is illustrated on some Southern Bluefin Tuna tagging and catch data. We also discuss important model assumptions and give suggestions for future research including the integration of tag-return and catch at age data analyses.     

Nonequilibrium coexistence in a competition model with nutrient storage

Resource competition theory predicts that, in equilibrium, the number of coexisting species cannot exceed the number of limiting resources. In some competition models, however, competitive interactions may result in nonequilibrium dynamics, allowing the coexistence of many species on few resources. The relevance of these findings is still unclear, since some assumptions of the underlying models are unrealistic. Most importantly, these models assume that individual growth directly reflects the availability of external resources, whereas real organisms can store resources, thereby decoupling their growth from external fluctuations. Here we study the effects of resource storage by extending the well-known Droop model to the context of multiple species and multiple resources. We demonstrate that the extended Droop model shows virtually the same complex dynamics as models without storage. Depending on the model parameters, one may obtain competitive exclusion, stable equilibrium coexistence, periodic and non-periodic oscillations, and chaos. Again, nonequilibrium dynamics allows for the coexistence of many species on few resources. We discuss our findings in the light of earlier work on resource competition, highlighting the role of luxury consumption, trade-offs in competitive abilities, and ecological stoichiometry.     

A spatial model of growth and competition strategies in coral communities

A discrete spatial simulation model is developed to investigate the type and intensity of biological and physical factors influencing the structure of coral communities. The model represents reproduction, growth, and interspecific competition by coral colonies in terms of “ownership” of space in a plot of reef habitat. Using data for several eastern Pacific coral species, the model reproduces observed changes in species composition and diversity during coral community development. Model results suggest that during early successional stages, or in areas that are frequently disturbed, larval colonization and rapid growth are more important than dominance achieved by extracoelenteric digestion or by growing over another coral in acquiring and maintaining possession of reef substrate. In mature communities that remain undisturbed, dominance is the best competitive strategy. Although the model was developed to study natural and man-induced changes in the community dynamics of coral reefs, it could be adapted to study other sessile organisms where spatial pattern is an important influence on the frequency and outcome of biological interactions.     

A general model for analyzing Taylor's spatial scaling laws

Taylor's spatial scaling law concerns the relation between the variance and the mean population counts within areas of a given size. For a range of area sizes, the log of the variance often is an approximately linear function of the mean with a slope between 1 and 2, depending on the range of areas considered. In this paper, we investigate this relationship theoretically for random quadrat samples within a large area. The model makes a distinction between the local point process determining the position of each individual and the population density described by a spatial covariance function. The local point process and the spatial covariance of population density both contribute to the general relationship between the mean and the variance in which the slope may begin at 1, increase to 2, and decrease to 1 again. It is demonstrated by an example that the slope theoretically may exceed 2 by a small amount for very regular patterns that generate spatial covariance functions that increase in certain intervals. We also show how properties of population dynamics in space and time determine this relationship.     

A general bioeconomic simulation model for annual-crop marine fisheries

A generalized bioeconomic simulation model of annual-crop marine fisheries is described and its use in marine fisheries management is demonstrated. The biological submodel represents the recruitment of new organisms into the fishery, the movement of organisms from one fishing area to another and from one depth to another, the growth of organisms and the mortality of organisms resulting both from natural causes and from fishing. The economic submodel represents the fishing effort exerted on each resource species, the monetary costs of fishing, the value of the harvest and the rent (or excess profits) to the fishery.Basic dynamics of the model results from changes in the number of organisms in the fishery over time, which can be summarized as a set of difference equations of the general form ΔN/Δt = R + I ? E ? M ? F where ΔN/Δt is the net change in number of organisms in the fishery over time, R is recruitment, I is immigration, E is emigration, M is natural mortality and F is fishing mortality. R is a driving variable, whereas I, E, M and F are functions of the state of the system at any given point in time. The model can be run in a deterministic or stochastic mode. Values for parameters affecting rates of recruitment, movement, growth, natural mortality and fishing mortality can be selected from uniform, triangular or normal distributions.Use of the model within a fisheries-management framework is demonstrated by evaluating several management alternatives for the pink shrimp (Penaeus duorarum) fishery on the Tortugas grounds in the Gulf of Mexico. Steps involved in use of the model, including parameterization, validation, sensitivity analysis and stochastic simulations of management policies, are explained.     

Stability and exploitation in two-species discrete time population models with delay

A two species model is developed which incorporates interactions at both the adult stage and the juvenile stage, as well as delays between birth and sexual maturity.The local stability of the model is examined. In particular, four necessary conditions are given and a simple sufficiency condition derived, which may avoid the need to calculate eigenvalues.The optimal steady state escapement levels are then derived under the assumption that neither species is completely removed. The optimal levels are those which maximize the discounted present value of the net economic yield.Finally, the results are applied to two models; one a familiar interspecific competition model and the other a two sex whaling model proposed by the author.     

A trait specific model of competition in a spatially structured plant community

The non-random dispersal of plant propagules is thought to counter competitive exclusion and thus promote the survival of competitively inferior species. We investigated this process by modelling the outcome of interactions between species with competitive ability defined as a function of both life-history traits and the environment with both random and clustered dispersal strategies and in environmentally homogeneous and heterogeneous environments. Four main results emerged: (1) environmental heterogeneity was seen to promote co-existence in conjunction with associated trait variation for tolerance to the environmental variable and where this trait variation was effectively limited by ‘trade-off’ such that no single species had an overall competitive advantage, (2) consistent with theory, random dispersal appeared to enhance the likelihood of competitive exclusion, whereas clustering favoured co-existence, (3) the ecological outcome of interactions between dispersal and competitive relationships varied as a function of the trait determining the competitive advantage within a particular environment, and (4) promotion of co-existence by clustered dispersal was most marked when associated with environmental heterogeneity. It is argued that these results suggest that current ecological models of species interactions may need to be modified to incorporate a more realistic understanding of competitive ability if we are to better understand the factors effecting species co-existence.     

A mathematical model of competition and colonization in a community of marine benthic algae

A mathematical model has been constructed for the algal community on the rocky shores of a Norwegian fjord. We report here on the studies of competition and colonization along a vertical transect from the upper intertidal to the sublittoral habiats. Results on species abundance and distribution (patterns of zonation) and time to reach maturity have been compared to observations both in the fjord area and in other rocky shore areas.Competition coefficients for the algae were inferred from plant morphology and shown to be in agreement with observations of algal abundance and their zone-forming ability. Competition restricts the distribution of the species, especially at the lower elevations, but does not alter their relative position. However, increasing uniform competition prolongs the time in which zone-forming can occur, and it also decreases the overall biomass which an area can sustain. Colonization by a single species may create transient stages in community development of the same order of magnitude as algae longevity, and probably also alters the zonation pattern to some degree.The simulation results indicate that the large-scale algal distribution pattern in the Hardangerfjord area results from global stability of the rocky shore community.     

Sustained dynamic transience in a Lotka-Volterra competition model system for grassland species

Theoretical approaches, such as the Lotka-Volterra framework, enable predictions about long term species coexistence based on stability criteria, but generally assume temporal constancy of system equations and parameters. In real world systems, temporal variability may interfere with the attainment of stable states. Managed grassland ecosystems in Northwestern Europe experience structural periodic fluctuations in environmental conditions: the seasons. In addition, periodic disturbances such as cutting are very common. Here we show, using a Lotka-Volterra system applied to grassland species with empirically derived parameters, that seasonal variability can result in a time dependent equilibrium and redirection of displacement processes.Parameter estimates differed between species and - in most cases - between the seasons. As a result, five of the fifteen tested species combinations had different outcomes of species interactions between seasons. This indicates that systems remain in dynamic transience over the year as the equilibrium changes and the species composition of the system follows the equilibrium without ever attaining it. The non-attainment of the steady state enables coexistence of species even if there is competitive exclusion in one of the seasons. For three of the fifteen species combinations, cutting frequency affected the long-term coexistence patterns. Cutting resets the biomass of competing species and favours during regrowth those species that have a high growth rate, which can alter species coexistence in comparison to a Lotka-Volterra model without cutting. The Lotka-Volterra framework with seasonally changing empirical parameters predicts coexistence as a possible outcome of systems that in component seasons are characterised by exclusion, and vice versa.     

Turbulence and phytoplankton diversity: A general model of the “paradox of plankton”

The principle of “competitive exclusion” predicts that no two species can occupy the same ecological niche at the same time and place (Hardin, 1960). Hutchinson (1953, 1961) suggested that the vast diversity of phytoplankton observed in many aquatic environments presents an apparent contradiction to this principle. Since all phytoplankton compete for the same basic resources, and since the euphotic zones of most natural waters are relatively homogeneous, such coexisting plankters appear to be simultaneously occupying the same niche. In this paper we present simulation results from a mathematical model wherein we examine the hypothesis that physical turbulence in an aquatic system can mollify interactive pressures between plankton populations and allow coexistence of species competing for the same resources. Using Bella's (1972) highly simplified model as a point of departure, we develop a new model, explicitly incorporating gross physiological mechanisms, to investigate the effects of both advective and turbulent components of water movement on the growth of three competing phytoplankton species. We observed that, in the absence of water motion, no two species were able to coexist, while under the hypothetical conditions of advection without turbulence (laminar flow), just two species were able to occur contemporaneously. Coexistence of all three species was achieved only with the addition of a random turbulent component to the model's hydrodynamic function. Moreover, this general coexistence was observed only when the major turbulent frequency approached the turnover rate of phytoplankton populations. We suggest that there is a limited region of periodicities and magnitudes for hydrodynamic energy in which phytoplankton can coexist, and that most natural aquatic environments fall within this region. We further speculate that, in general, the coupling of physical and biological processes in nature may be influenced by the relative frequency characteristics of those processes.     

Vegetation competition model for water and light limitation. I: Model description, one-dimensional competition and the influence of groundwater

Vegetation growth models often concentrate on the interaction of vegetation with soil moisture but usually omit the influence of groundwater. However the proximity of groundwater can have a profound effect on vegetation growth, because it strongly influences the spatial and temporal distribution of soil moisture and therefore water and oxygen stress of vegetation. In two papers we describe the behavior of a coupled vegetation-groundwater-soil water model including the competition for water and light. In this first paper we describe the vegetation model, compare the model to measured flux data and show the influence of water and light competition in one dimension. In the second paper we focus on the influence of lateral groundwater flow and spatial patterns along a hillslope. The vegetation model is based on a biophysical representation of the soil-plant-atmosphere continuum. Transpiration and stomatal conductance depend both on atmospheric forcing and soil moisture content. Carbon assimilation depends on environmental conditions, stomatal conductance and biochemical processes. Light competition is driven by tree height and water competition is driven by root water uptake and its water and oxygen stress reaction. The modeled and measured H₂O and CO₂ fluxes compare well to observations on both a diurnal and a yearly timescale. Using an upscaling procedure long simulation runs were performed. These show the importance of light competition in temperate forests: once a tree is established under slightly unfavorable soil moisture conditions it can not be outcompeted by smaller trees with better soil moisture uptake capabilities, both in dry as in wet conditions. Performing the long simulation runs with a background mortality rate reproduces realistic densities of wet and dry adapted tree species along a wet to dry gradient. These simulations show that the influence of groundwater is apparent for a large range of groundwater depths, by both capillary rise and water logging. They also show that species composition and biomass have a larger influence on the water balance in eco-hydrological systems than soil and groundwater alone.     

Pollution control is a two-sector dynamic general equilibrium model

This paper examines the problem of pollution control is an economy which produces two commodities which have different polluting characteristics. The model sheds light on the change is the optimal combination of commodities such as public transportation as opposed to private transportation, detergents and soap, returnable and nonreturnable bottles, and so on.     

A dynamic model of mating behavior in digger wasps: the energetics of male-male competition mimic size-dependent thermal constraints

I develop a state-based dynamic model of behavior to demonstrate that size-dependent differences in temperature tolerances are not necessary to account for the activity of small male digger wasps late in the day. In the model, males defend or patrol the nesting area, wait near nests, or feed away from the nesting area depending on time of day, energy reserves and size rank. I assume a large male competitive advantage, so mating opportunities decrease with size rank for territorial or patrolling males and are rare for all waiting males; the costs of patrolling or defense are higher than the costs of waiting. If energy reserves of all males are initially small, all males alternate feeding and territorial or patrolling behavior. If energy reserves are initially large, large males patrol or maintain territories until they risk starvation and leave the area to feed. At this time, smaller males that have conserved their resources by waiting and feeding may defend territories or patrol. I simulate the behavior of three populations representing two species of Microbembex by assuming large initial energy reserves for populations in which males were territorial and small initial reserves for populations in which males patrolled, and then convert the predicted time of activity to temperature using local regressions from field studies. Temporal patterns in the activity of large and small males were similar to those actually observed, and relationships between size and temperature predicted by the model corresponded to most observations and were sometimes positive. Thus, the delayed activity of smaller males does not correspond to activity at higher temperatures and is probably not attributable to size-dependent thermal tolerances, but may represent a temporal displacement of mating activity due to intra-sexual competition and mediated by energetics. The model makes testable predictions on the timing of feeding and depletion of energy reserves in relation to size and initial energy state, and suggests how differences among species may influence the temporal and spatial organization of male mating behavior. Received: 27 February 1997 / Accepted after revision: 26 July 1997     

Implications of a simple competition model for the stability of an intercropping system

It is a common view that intercropping systems of agricultural crops produce more stable yields than do systems in which the same crops are grown in monoculture. This paper discusses a modelling approach which has been used to support the notion that whether or not intercropping is more stable than monoculture depends on the mode of interaction among crops, i.e. whether two different crops suppress or enhance each other. It is shown here that this notion is not supported by the model used. We conclude that the relative merits of the two cropping systems depend on the proportion of land allocated to each crop rather than on the mode of interaction. The model suggests that if the optimum allocation of land is considered, both systems will be equally stable.     

A general random walk model for the leptokurtic distribution of organism movement: Theory and application

The movement of organisms is usually leptokurtic in which some individuals move long distances while the majority remains at or near the area they are released. There has been extensive research into the origin of such leptokurtic movement, but one important aspect that has been overlooked is that the foraging behaviour of most organisms is not Brownian as assumed in most existing models. In this paper we show that such non-Brownian foraging indeed gives rise to leptokurtic distribution. We first present a general random walk model to describe the organism movement by breaking the foraging of each individual into events of active movement and inactive stationary period; its foraging behaviour is therefore fully characterized by a joint probability of how far the individual can move in each active movement and the duration it remains stationary between two consecutive movements. The spatio-temporal distribution of the organism can be described by a generalized partial differential equation, and the leptokurtic distribution is a special case when the stationary period is not exponentially distributed. Empirical observations of some organisms living in different habitats indicated that their rest time shows a power-law distribution, and we speculate that this is general for other organisms. This leads to a fractional diffusion equation with three parameters to characterize the distributions of stationary period and movement distance. A method to estimate the parameters from empirical data is given, and we apply the model to simulate the movement of two organisms living in different habitats: a stream fish (Cyprinidae: Nocomis leptocephalus) in water, and a root-feeding weevil, Sitona lepidus in the soil. Comparison of the simulations with the measured data shows close agreement. This has an important implication in ecology that the leptokurtic distribution observed at population level does not necessarily mean population heterogeneity as most existing models suggested, in which the population consists of different phenotypes; instead, a homogeneous population moving in homogeneous habitat can also lead to leptokurtic distribution.     

Rank differences in energy intake rates in white-faced capuchin monkeys, Cebus capucinus: the effects of contest competition

The effect of aggressive competition over food resources on energy intake rate is analyzed for individuals of three groups of 25–35 white-faced capuchin monkeys, Cebus capucinus, living in and near Lomas Barbudal Biological Reserve, Costa Rica. An individuals energy intake rate on a given food species was affected by its rank and the number of agonistic interactions within the feeding tree. Dominant group members had higher energy intake rates relative to subordinate group members whether or not there was agonism within the feeding tree. Low- and mid-ranked individuals had lower energy intake rates in trees with higher amounts of aggression, while energy intake rate of high-ranked individuals was not affected by the amount of aggression in the feeding tree. Energy intake was not influenced by the sex of the individual when rank was held constant statistically. Energy intake was positively correlated with total crown energy (measured in kilojoules) within the feeding tree for two of three study groups. This difference may be explained by the quality of each groups territory. Finally, high-ranked individuals are responsible for the majority of agonism within feeding trees and target middle- and low-ranked individuals equally. These findings fit the predictions of current socioecological models for within-group contest competition over food resources. The results of this study suggest that within-group competition affects energy intake rate in white-faced capuchin monkeys.     

Coupling soil water and shoot dynamics in three grass species: A spatial stochastic model on water competition in Neotropical savanna

Savannas are ecosystems known for their high environmental and economic value. They cover at least 20% of the global land surface and, in some cases, can act as a boundary between tropical rainforest and deserts. Water is an important determinant of savanna ecosystems.In this paper, we present a theoretical stochastic model of root competition for water, which couples, soil water availability, phenology, and root and shoot architecture applied to three Neotropical savanna grasses. Soil moisture was simulated using a daily balance, as proposed by Rodriguez-Iturbe et al. [Rodriguez-Iturbe, I., Porporato, A., Ridolfi, L., Isham, V., Cox, D.R., 1999. Probabilistic modelling of water balance at a point: the role of climate, soil and vegetation. Proc. R. Soc. London, Ser. A 455, 3789–3805.]. To simulate rainfall stochasticity, we used daily precipitation data from the airport weather station in the State of Barinas, Venezuela, for the period 1991–2007. Competition among neighbouring plants took into account the spatial distribution of the individuals. As a final step, the model allowed us to calculate the shoot dynamic of the species as a function of soil water availability.Using these data, we compared the behaviour of isolated plants, pairs and trios, and we found below-ground competition to be a fundamental component of global (shoot + root) competition. Finally, our model suggests various circumstances that allow poor competitor plants to coexist in competition for water with more successful competitors. Apparently, this is not only due to transpiration rates, but also to differences in shoot emergence and shoot growth.