Mathematical Ecology

Honey bees (Apis mellifera) are responsible for pollinating nearly 80% of all food-producing plants, therefore, the success or failure of a colony is vital to global food production. However, Vector-borne diseases account for approximately the 20% of all infectious diseases leading to the use of insecticides to control them. These chemical products not only affect mosquitoes but other insects such as Honey bees, causing imbalances across most ecosystems around the world. Our goal is to describe the effect of insecticides over brood death rate and its possible effects over Honey Bee population dynamics. For this, we constructed a compartmental model using ordinary differential equations (ODE's) to describe brood birth, mortality, maturation and adult dynamics. To analyze our model we solved the equilibriums and performed stability and sensitivity analysis. We used data from 1975-76 in order to see how well our model fits the data and estimate the parameters. Besides, we implement...

"""Lyapunov functions for some mathematical models of ecological commensalism between two species are introduced. Global stability of the unique positive equilibrium is thereby established.

Keyword: Models of ecological commensalism; Lyapunov's asymptotic stability theorem; Global stability"""

This paper addresses the global stability of a two-species Lotka-Volterra cooperative system with four discrete delays via the method of Lyapunov functionals. We apply our Lyapunov functional techniques to prove stability of Lotka-Volterra models of mutualism with two delays or without delays. We note that, if all the delayed feedbacks are positive then the stability properties are independent of the values of the delays. We compare our results with known results.

Keywords: Lotka-Volterra cooperative system, discrete delays, delayed positive feedback, Lyapunov functional, global stability.

Plant species vary greatly in their responsiveness to nutritional soil mutualists, such as my- corrhizal fungi and rhizobia, and this responsiveness is associated with a trade-off in alloca- tion to root structures for resource uptake. As a result, the outcome of plant competition can change with the density of mutualists, with microbe-responsive plant species having high competitive ability when mutualists are abundant and non-responsive plants having high competitive ability with low densities of mutualists. When responsive plant species also allow mutualists to grow to greater densities, changes in mutualist density can generate a positive feedback, reinforcing an initial advantage to either plant type. We study a model of mutualist-mediated competition to understand outcomes of plant-plant interactions within a patchy environment. We find that a microbe-responsive plant can exclude a non-responsive plant from some initial conditions, but it must do so across the landscape including in the mi- crobe-free areas where it is a poorer competitor. Otherwise, the non-responsive plant will persist in both mutualist-free and mutualist-rich regions. We apply our general findings to two different biological scenarios: invasion of a non-responsive plant into an established mi- crobe-responsive native population, and successional replacement of non-responders by microbe-responsive species. We find that resistance to invasion is greatest when seed dis- persal by the native plant is modest and dispersal by the invader is greater. Nonetheless, a native plant that relies on microbial mutualists for competitive dominance may be particular- ly vulnerable to invasion because any disturbance that temporarily reduces its density or that of the mutualist creates a window for a non-responsive invader to establish dominance. We further find that the positive feedbacks from associations with beneficial soil microbes create resistance to successional turnover. Our theoretical results constitute an important first step toward developing a general understanding of the interplay between mutualism and competition in patchy landscapes, and generate qualitative predictions that may be tested in future empirical studies.

In this paper an eco-epidemiological model incorporating a prey refuge and prey harvesting with disease in the prey-population is considered. Predators are assumed to consume both the susceptible and infected prey at different rates. The positivity and boundedness of the solution of the system are discussed. The existence and stability of the biologically feasible equilibrium points are investigated. Numerical simulations are performed to support our analytical findings.

This article concerns a systemic manifestation of small scale interfacial heterogeneities in large scale quantities of interest to a variety of diverse applications spanning the earth, biological and ecological sciences. Beginning with formulations in terms of partial differential equations governing the conservative, advective-dispersive transport of mass concentrations in divergence form, the specific interfacial heterogeneities are introduced in terms of (spatial) discontinuities in the diffusion coefficient across a lower-dimensional hypersurface. A pathway to an equivalent stochastic formulation is then developed with special attention to the interfacial effects in various functionals such as first passage times, occupation times and local times. That an appreciable theory is achievable within a framework of applications involving one-dimensional models having piecewise constant coefficients greatly facilitates our goal of a gentle introduction to some rather dramatic mathematical consequences of interfacial effects that can be used to predict structure and to inform modeling.

Modern day agriculture is dependent on the use of chemical agents for maximizing crop yield. This practice has perilous effects on the ecosystem if used in an unchecked manner. The present paper develops a mathematical model of an agricultural system that incorporates the cumulative effect of the chemical agents like pesticides, fertilizers etc. Dynamical behaviour of the system, such as boundedness, permanence and stability, are studied. Numerical simulations are carried out to study the changes in the behaviour of the system due to varying levels of potency of the chemicals. Based on these, an effort is made to determine the conditions necessary for a sustainable and productive agricultural system in presence of chemical agents. Applicability of the model in related scenarios is also discussed.

We develop a stochastic spatial model for Aedes aegypti populations based on the life cycle of the mosquito and its dispersal. Our validation corresponds to a monitoring study performed in Buenos Aires. Lacking information with regard to the number of breeding sites per block, the corresponding parameter (BS) was adjusted to the data. The model is able to produce numerical data in very good agreement with field results during most of the year, the exception being the fall season. Possible causes of the disagreement are discussed. We analyzed the mosquito dispersal as an advantageous strategy of persistence in the city and simulated the dispersal of females from a source to the surroundings along a 3-year period observing that several processes occur simultaneously: local extinctions, recolonization processes (resulting from flight and the oviposition performed by flyers), and colonization processes resulting from the persistence of eggs during the winter season. In view of this process, we suggest that eradication campaigns in temperate climates should be performed during the winter time for higher efficiency.

As we know from theory of evolution in ecology that the generalist predators feed on a variety of prey species may tend to persist in an ecosystem even if one particular prey species is absent in that environment. Predation by generalist predators is followed by a sigmoidal functional response which is ensured in literature. We already know that a mathematical predator–prey type model with sigmoidal functional response is found to be more realistic and stable than for a Holling type II functional response. In our model the generalist predator may go to extinct in the absence of one focal prey. It’s dynamics are studied in terms of the local and global aspect and stabilities are verified
around the positive equilibrium with the impact of constant delay in the functional response term as some time will be required for gestation of prey for predators. The condition for Hopf-bifurcation of the system has been derived. Numerical computer simulations have been done and compared with the analytical results of the model analysis.

We propose and analyze a mathematical model in ordinary differential equations to describe the dynamics of mosquitoes infested by bacteria. The introduction of some bacteria in mosquitoes population aims to diminish gradually the transmission of vector host-diseases. This is a good strategy of biological control.

Species coexistence involving trophic interactions has been investigated under two theoretical frameworks—partitioning shared resources and accessing exclusive resources. The influence of body size on coexistence is well studied under the exclusive resources framework, but has received less attention under the shared-resources framework. We investigate body-size-dependent allometric extensions of a classical MacArthur-type model where two consumers compete for two shared resources. The equilibrium coexistence criteria are compared against the general predictions of the alternative framework over exclusive resources. From the asymmetry in body size allometry of resource encounter versus demand our model shows, counterintuitively, and contrary to the exclusive resource framework, that a smaller consumer should be competitively superior across a wide range of supplies of the two resource types. Experimental studies are reviewed to resolve this difference among the two frameworks that arise from their respective assumptions over resource distribution. Another prediction is that the smaller consumer may have relatively stronger control over equilibrium resource abundance, and the loss of smaller consumers from a community may induce relatively stronger trophic cascades. Finally, from satiating consumers’ functional response, our model predicts that greater difference among resource sizes can allow a broader range of consumer body sizes to coexist, and this is consistent with the predictions of the alternative framework over exclusive resources. Overall, this analysis provides an objective comparison of the two alternative approaches to understand species coexistence that have heretofore developed in relative isolation. It advances classical consumer–resource theory to show how body size can be an important factor in resource competition and coexistence.

We propose and analyze a mathematical model in ordinary differential equations to describe the dynamics of mosquitoes infested by bacteria. The introduction of some bacteria in mosquitoes population aims to diminish gradually the transmission of vector host-diseases. This is a good strategy of biological control.

A three-species ecological model is proposed that comprises one prey and two predators (low predator and top predator) to understand the persistence and/or extinction of prey and predators under a suitable environment. The realistic ecological system is typically inhomogeneous in terms of spatiotemporal variations of population density. With this motivation, white noise is introduced to the system to conceive the impact of environmental heterogeneity on the interaction of prey and predators. The boundedness of solutions of the system is presented. The condition for the local stability of the deterministic model is established. The stochastic analysis of the system suggests the impression of the environmental noise on the dynamical behaviour of solutions. Numerical results reveal that the proposed system exhibits an oscillatory pattern of solutions. It is shown that the white noise in the low predator has a dominating role in comparison with other noises to keep the oscillation in order. Moreover, to investigate the spatial dependency of the population density, diffusion term is introduced to the system that also displays oscillatory behaviour of the solutions. The results also confirm that the population density of prey, low predator, and top predator have a biological control in the oscillatory behaviour. In the presence of diffusion and white noise, numerical simulation illustrates the sensitivity of the diffusion coefficients and the amplitude of noises. Numerical results lead to a reasonable agreement with the analytical findings in both non-spatial and spatial systems.

The study analyzed sustainable agricultural resource use in Nigeria using the Ecological Footprint Approach. The results showed that the emergy density of the earth and the country were 3.10E+10 sej/m 2 /year and 1.33E+11 sej/m 2 /year respectively, indicating that a total energy of 3.10E+10 sej was used up per meter square of the total global hectares to provide for the ecosystem services. Given the interaction of the renewable resource flows a total energy of 1.33E+11sej was expended in the transformation of agricultural resources per meter square of the land area of the country. The demand for resources per capita were 5.25ha of arable land, 2.54 ha of pasture land, 0.816ha of water area, 0.131 ha of forest land, 2.6ha of fossil land and 0.0000481 ha of built-up area with a total ecological footprint of 11.3 hectares per capita; while carrying capacity per capita was 5.2 hectares, showing a deficit of-6.1 hectares per capita and indicating unsustainable agricultural resource use-hence sustainable development cannot be achieved. The study however recommended that ecological farming, organic farming and other sustainable agricultural systems that reduce the footprint for arable land should be adopted in the country, and that Nigeria should strategize ways of reducing the importation and consumption of foreign food crops and encourage local food crop farmers – as this will reduce the ecological footprint of the country, and to ensure
that agricultural resources are sustainable every citizen of the country should adjust his/her lifestyle
so as to reduce consumption by 6.1 hectares per person, preserve and conserve agricultural
resources for the current use and the future generation – as this will not only secure sustainable
use of agricultural resources but pave ways to achieving sustainable development in the country

We develop a stochastic spatial model for Aedes aegypti populations based on the life cycle of the mosquito and its dispersal. Our validation corresponds to a monitoring study performed in Buenos Aires. Lacking information with regard to the number of breeding sites per block, the corresponding parameter (BS) was adjusted to the data. The model is able to produce numerical data in very good agreement with field results during most of the year, the exception being the fall season. Possible causes of the disagreement are discussed. We analyzed the mosquito dispersal as an advantageous strategy of persistence in the city and simulated the dispersal of females from a source to the surroundings along a 3-year period observing that several processes occur simultaneously: local extinctions, recolonization processes (resulting from flight and the oviposition performed by flyers), and colonization processes resulting from the persistence of eggs during the winter season. In view of this process, we suggest that eradication campaigns in temperate climates should be performed during the winter time for higher efficiency.

In a recent paper, critique the use of Le´vy walks to describe animal movement, arguing that non-Le´vy walk processes could be misidentified as Le´vy patterns and, conversely, movement patterns actually generated by Le´vy processes may be wrongly attributed to other mechanisms. The authors suggest that this ambiguity is partly caused by sampling paths at scales that do not reflect actual movement decisions and this despite the theoretical scale-independence of Le´vy walks. These findings, if true, would be an important contribution, as the Le´vy walk is a popular, although controversial, model in the animal movement literature.

"""Lyapunov functions for two-species mutualism models subject to proportional harvesting are introduced. Global stability of the unique coexisting equilibrium is thereby established.

Keywords: Models of facultative mutualism; Harvesting; Lyapunov’s asymptotic stability theorem; Global stability."""

The broad objective of this study was to analyse sustainable agricultural resource use in Nigeria using the Ecological Footprint Approach. The specific objectives included to; estimate the global emergy and emergy density, determine the ecological footprint per capita, analyse the per capita bio-capacity, and ascertain the level of agricultural resource use, sustainability status and resource substitution in Nigeria. The total land area, population and thematic data of the country were used for the study.

We investigate the dynamical behaviors of two-prey one-predator model with general Holling type functional responses. The effect of seasonal perturbation on the model has been discussed analytically as well as numerically. The periodic fluctuation is considered in prey growth rate and the predator mortality rate of the model. The impulsive effects involving biological and chemical control strategy, periodic releasing of natural enemies, and spraying pesticide at different fixed times are introduced in the model with seasonal perturbation. We derive the conditions of stability for impulsive system using Floquet theory, small amplitude perturbation skills. A local asymptotically stable prey (pest) eradicated periodic solution is obtained when the impulsive period is less than some critical value. Numerical simulations of the model with and without seasonal disturbances exhibit different dynamics. Also we simulate numerically the model involving seasonal perturbations without impulse a...

Random walks are used to model movement in a wide variety of contexts: from the movement of cells undergoing chemotaxis to the migration of animals. In a twodimensional biased random walk, the diffusion about the mean drift position is entirely dependent on the moments of the angular distribution used to determine the movement direction at each step. Here we consider biased random walks using several different angular distributions and derive expressions for the diffusion coefficients in each direction based on either a fixed or variable movement speed, and we use these to generate a probability density function for the long-time spatial distribution. We demonstrate how diffusion is typically anisotropic around the mean drift position and illustrate these theoretical results using computer simulations. We relate these results to earlier studies of swimming microorganisms and explain how the results can be generalized to other types of animal movement.

Species coexistence involving trophic interactions has been investigated under two theoretical frameworks-partitioning shared resources and accessing exclusive resources. The influence of body size on coexistence is well studied under the exclusive resources framework, but has received less attention under the sharedresources framework. We investigate body-size-dependent allometric extensions of a classical MacArthur-type model where two consumers compete for two shared resources. The equilibrium coexistence criteria are compared against the general predictions of the alternative framework over exclusive resources. From the asymmetry in body size allometry of resource encounter versus demand our model shows, counterintuitively, and contrary to the exclusive resource framework, that a smaller consumer should be competitively superior across a wide range of supplies of the two resource types. Experimental studies are reviewed to resolve this difference among the two frameworks that arise from their respective assumptions over resource distribution. Another prediction is that the smaller consumer may have relatively stronger control over equilibrium resource abundance, and the loss of smaller consumers from a community may induce relatively stronger trophic cascades. Finally, from satiating consumers' functional response, our model predicts that greater difference among resource sizes can allow a broader range of consumer body sizes to coexist, and this is consistent with the predictions of the alternative framework over exclusive resources. Overall, this analysis provides an objective comparison of the two alternative approaches to understand species coexistence that have heretofore developed in relative isolation. It advances classical consumer-resource theory to show how body size can be an important factor in resource competition and coexistence.

Honey bees (Apis mellifera) are responsible for pollinating nearly 80% of all food-producing plants, therefore, the success or failure of a colony is vital to global food production. However, not all insects are beneficial to humankind and sometimes it is necessary to reduce the population of disease-carrying mosquitoes by using insecticides. These chemical products not only affect mosquitoes but other insects such as Honey bees, causing imbalances across most ecosystems around the world. Our goal is to describe the influence of brood deaths on Honey Bee population dynamics and the potential impact of insecticides on these natural changes. We constructed a compartmental model using ordinary differential equations to describe brood birth, mortality, maturation and how the newly emerged adults live and die. To obtain our parameters we used data from 1975-76 to estimate the parameters in order to see how well our model fits the data. Finally, we implemented seasonality in our simulatio...

Certain major topics on the subject of interregional and international migration of population stocks, as well as of the interregional and international flows of the two central input factors in the economic production process – namely capital and labor – are addressed in this paper. The key findings are that: first, uncontrolled global migration inevitably leads to a uniform utility level equal to the globally prevailing average: that of the average Indian and Chinese peasants. Second, that uncontrolled flows of capital and labor result in dynamically chaotic urban, regional and national economic structures. This paper revisits, summarizes and extends a number of papers (with a focus on three of them) by this author, written and published in the period 1975 – 1985.

We develop a stochastic spatial model for Aedes aegypti populations based on the life cycle of the mosquito and its dispersal. Our validation corresponds to a monitoring study performed in Buenos Aires. Lacking information with regard to the number of breeding sites per block, the corresponding parameter (BS) was adjusted to the data. The model is able to produce numerical data in very good agreement with field results during most of the year, the exception being the fall season. Possible causes of the disagreement are discussed. We analyzed the mosquito dispersal as an advantageous strategy of persistence in the city and simulated the dispersal of females from a source to the surroundings along a 3-year period observing that several processes occur simultaneously: local extinctions, recolonization processes (resulting from flight and the oviposition performed by flyers), and colonization processes resulting from the persistence of eggs during the winter season. In view of this process, we suggest that eradication campaigns in temperate climates should be performed during the winter time for higher efficiency.

We propose two models of one hyper-connected mutualistic-species  described by delay differential equations of Lotka-Volterra type. An hyper-connected model comprises a central species interacting with a number of peripheral species around it, that is to say,  one animal species (pollinators or dispersers) that interacts with several plant species (flowering plants or fruit trees), or several animal species that interact with one plant species. We derive a necessary and sufficient condition for the global asymptotic stability of the unique coexisting steady state of hyper-connected systems by means of novel Lyapunov functionals.

Keywords: Lotka--Volterra cooperative  system; discrete delays; distributed delays;  Lyapunov functional; global stability.

Necessary conditions for coexistence of competitive species described by a simple model of population growth and resource exploitation are derived using controllability and observability conditions for time-varying linear systems. The “competitive exclusion principle” of ecological theory follows from these

The main objective of this work is to determine the conditions for coexistence and competitive exclusion in a discrete model for a community of three species: a stage-structured host and two competing parasitoids sharing the same host developmental stage.