Lotka-Volterra Type Model Research Articles

Dynamics of a Lotka-Volterra type model with applications to marine phage population dynamics

The famous Lotka-Volterra equations play a fundamental role in the mathematical modeling of various ecological and chemical systems. A new modification of these equations has been recently suggested to model the structure of marine phage populations, which are the most abundant biological entities in the biosphere. The purpose of the paper is: (i) to make some methodical remarks concerning this modification; (ii) to discuss new types of canards which arise naturally in this context; (iii) to present results of some numerical experiments.

Remarks on some Lotka–Volterra type cross-diffusion models

The present article collects some results for Lotka–Volterra type models in the presence of cross-diffusion effects. It discusses the existence and the nonexistence of nonhomogeneous steady state solutions and deals with the existence of traveling wave solutions in the presence of cross-diffusion effects.

Contractivity and global stability for discrete models of Lotka-Volterra type

Consider the following discrete model of nonautonomous Lotka-Volterra type: Formula math. where Formula math. In this paper, to the above discrete system, we establish two type sufficient conditions that ensure the of solutions which are sufficient conditions of the global asymptotic stability of system. This is an extension of the former work (2002, J. Math. Anal. Appl. 270. 602-635) for n = 1 to n ≥ 2. In particular, for autonomous case of the above system, the sufficient condition for the global asymptotic stability of the positive equilibrium offered by Wang et al. (2002, J. Math. Anal. Appl. 264, 147-169), is proved to satisfy this contractivity of the positive equilibrium.

Modelling allelopathy among marine algae

Allelopathy among microalgae has been modelled in order to fit the dynamics of mixed cultures of Chrysocromulina polylepis and Heterocapsa triquetra, carried out with three different initial concentrations of the allelopathic species C. polylepis. The experimental data have been analysed with a simple Lotka–Volterra type model including an allelopathic term. Based on the population dynamics of the two species in unialgal and mixed cultures, the model has been simplified and solved analytically. The best fit of the allelopathy parameter ( γ ≈ 1 0 − 5 to 1 0 − 6 ) provides a good agreement between the theoretical curve and the experimental data for the two highest initial concentrations of C. polylepis but, interestingly, shows a less accurate fit for low initial concentrations of the toxic alga. Following the same procedures, a modified model, in which the allelopathic effect is dependent on the square of the concentration of the toxic alga, has been adjusted to the experimental data. This modified model presents a good fit for all the range of initial concentrations of the toxic alga. These results allow a quantification of the strength of the allelopathic interaction between two marine phytoplankton species. The growth curve of the non-toxic alga is significantly affected by the allelopathy term only after the toxic alga has reached relatively high concentrations. This supports field observations suggesting that, at least in the case of C. polylepis, allelopathy may be important once a bloom is well developed, but is not likely to be a key factor in initial phases of the proliferation.

Consequences of symbiosis for food web dynamics.

Basic Lotka-Volterra type models in which mutualism (a type of symbiosis where the two populations benefit both) is taken into account, may give unbounded solutions. We exclude such behaviour using explicit mass balances and study the consequences of symbiosis for the long-term dynamic behaviour of a three species system, two prey and one predator species in the chemostat. We compose a theoretical food web where a predator feeds on two prey species that have a symbiotic relationships. In addition to a species-specific resource, the two prey populations consume the products of the partner population as well. In turn, a common predator forages on these prey populations. The temporal change in the biomass and the nutrient densities in the reactor is described by ordinary differential equations (ODE). Since products are recycled, the dynamics of these abiotic materials must be taken into account as well, and they are described by odes in a similar way as the abiotic nutrients. We use numerical bifurcation analysis to assess the long-term dynamic behaviour for varying degrees of symbiosis. Attractors can be equilibria, limit cycles and chaotic attractors depending on the control parameters of the chemostat reactor. These control parameters that can be experimentally manipulated are the nutrient density of the inflow medium and the dilution rate. Bifurcation diagrams for the three species web with a facultative symbiotic association between the two prey populations, are similar to that of a bi-trophic food chain; nutrient enrichment leads to oscillatory behaviour. Predation combined with obligatory symbiotic prey-interactions has a stabilizing effect, that is, there is stable coexistence in a larger part of the parameter space than for a bi-trophic food chain. However, combined with a large growth rate of the predator, the food web can persist only in a relatively small region of the parameter space. Then, two zero-pair bifurcation points are the organizing centers. In each of these points, in addition to a tangent, transcritical and Hopf bifurcation a global heteroclinic bifurcation is emanating. This heteroclinic cycle connects two saddle equilibria where the predator is absent. Under parameter variation the period of the stable limit cycle goes to infinity and the cycle tends to the heteroclinic cycle. At this global bifurcation point this cycle breaks and the boundary of the basin of attraction disappears abruptly because the separatrix disappears together with the cycle. As a result, it becomes possible that a stable two-nutrient-two-prey population system becomes unstable by invasion of a predator and eventually the predator goes extinct together with the two prey populations, that is, the complete food web is destroyed. This is a form of over-exploitation by the predator population of the two symbiotic prey populations. When obligatory symbiotic prey-interactions are modelled with Liebig's minimum law, where growth is limited by the most limiting resource, more complicated types of bifurcations are found. This results from the fact that the Jacobian matrix changes discontinuously with respect to a varying parameter when another resource becomes most limiting.

Stoichiometry in Producer–Grazer Systems: Linking Energy Flow with Element Cycling

All organisms are composed of multiple chemical elements such as carbon, nitrogen and phosphorus. Whileenergy flow and element cycling are two fundamental and unifying principles in ecosystem theory, population models usually ignore the latter. Such models implicitly assume chemical homogeneity of all trophic levels by concentrating on a single constituent, generally an equivalent of energy. In this paper, we examine ramifications of an explicit assumption that both producer and grazer are composed of two essential elements: carbon and phosphorous. Using stoichiometric principles, we construct a two-dimensional Lotka–Volterra type model that incorporates chemical heterogeneity of the first two trophic levels of a food chain. The analysis shows that indirect competition between two populations for phosphorus can shift predator–prey interactions from a (+, −) type to an unusual (−, −) class. This leads to complex dynamics with multiple positive equilibria, where bistability and deterministic extinction of the grazer are possible. We derive simple graphical tests for the local stability of all equilibria and show that system dynamics are confined to a bounded region. Numerical simulations supported by qualitative analysis reveal that Rosenzweig’s paradox of enrichment holds only in the part of the phase plane where the grazer is energy limited; a new phenomenon, the paradox of energy enrichment, arises in the other part, where the grazer is phosphorus limited. A bifurcation diagram shows that energy enrichment of producer–grazer systems differs radically from nutrient enrichment. Hence, expressing producer–grazer interactions in stoichiometrically realistic terms reveals qualitatively new dynamical behavior.

Implications of Spatial Heterogeneity for the Paradox of Enrichment

We analyze a simple model to show that spatial heterogeneity of zooplankton can explain discrepancies between the behavior of classical predator—prey models and the patterns observed in natural planktonic systems. We use a Lotka—Volterra type model of Daphia and algae. Daphnia occupies only a part of total volumes whereas the algae grow in the entire volume and diffuse between the two compartments. This simple spatial structure suffices to explain the observations that (1) natural Daphnia—algae systems tend to be relatively stable up to high nutrient values, and that (2) in the presence of Daphnia edible algae do increase with enrichment. Additionally, the model trivially explains confusing observations of oscillating Daphnia densities in the presence of a practically constant density of edible algae. The model is supported by the results of a laboratory experiment with a cascade of zooplankton—phytoplankton containers, devised originally to test ratio—dependent foraging. We derive minimalizations of our model, which no longer explicitly account for the spatial structure, but still preserve the essential behavior of the full model.

A Simplified Gypsy Moth Model System: model definition and description

A simple discrete Lotka-Volterra type model of the gypsy moth (Lymantria dispar L.) was developed using the more detailed Gypsy Moth Life System Model and related information. The tree equation is derived from the diameter growth of a gap model, extended to include crowding effects of stocking. The gypsy moth equation tracks eggs from generation to generation and translates this to subsequent larval density to predict food requirements of the population each year. The model explores the effect of gypsy moth feeding on diameter growth of trees and the available foliage on population dynamics. This model exhibits an exceptional range of behaviors. We use analytic techniques to explore the bases for these behaviors.

Effects of diffusion in fluctuating media: A noise-induced phase transition

An example of a noise-induced phase transition in a chemical system with diffusion is studied. It is shown that effects of diffusion can radically change the classical results on extinction and coexistence in the models of Lotka-Volterra type. Fluctuation phenomena near the point of the noise-induced phase transition are discussed.

Persistence in food chains with general interactions

Sufficient conditions for persistence in n-link food chain models with interaction functions g i are given: for g i nondecreasing and n = 3, and for g i positive in some compact interval to the right of zero and general n. The results generalize those for Lotka-Volterra type models. Necessary conditions for persistence are also given.