Population interactions modelled by bond graphs
Dionysis Assimacopoulos1986, Applied Mathematical Modelling
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
Bond graph analysis is applied to a category of ecosystem which examines trophic interactions between populations. A bond graph model for a trophic food chain is analysed, and correlation to Lotka-Volterra dynamics reveals the nature of all the conjugate variables and permits the causeeffect relationships of the mass-energy transformations to be model led explicitly. Other models are considered, as well as thermodynamic restrictions imposed by the graph topology of the systems.
Related papers
A general approach to the modelling of trophic chains
Ecological modelling, 2000
Based on the law of mass action (and its microscopic foundation) and mass conservation, we present here a method to derive consistent dynamic models for the time evolution of systems with an arbitrary number of species. Equations are derived through a mechanistic description, ensuring that all parameters have ecological meaning. After discussing the biological mechanisms associated to the logistic and Lotka-Volterra equations, we show how to derive general models for trophic chains, including the effects of internal states at fast time scales. We show that conformity with the mass action law leads to different functional forms for the Lotka-Volterra and trophic chain models. We use mass conservation to recover the concept of carrying capacity for an arbitrary food chain.
POPULATION DYNAMICS ON COMPLEX FOOD WEBS
Advances in Complex Systems, 2011
In this work we analyse the topological and dynamical properties of a simple model of complex food webs, namely the niche model. In order to underline competition among species, we introduce "prey" and "predators" weighted overlap graphs derived from the niche model and compare synthetic food webs with real data. Doing so, we find new tests for the goodness of synthetic food web models and indicate a possible direction of improvement for existing ones. We then exploit the weighted overlap graphs to define a competition kernel for Lotka-Volterra population dynamics and find that for such a model the stability of food webs decreases with its ecological complexity.
TOPOLOGICAL ENERGY DYNAMICS OF FOOD WEB GRAPHS
This paper explores the directed forest complex of food web graphs, which model the flow of energy in ecosystems. By applying discrete Morse theory, we construct near-perfect discrete Morse vector fields on the directed forest complex and provide necessary conditions for these fields to be perfect. Our results reveal deep connections between the combinatorial structure of food web graphs and their topological properties.
Consequences of symbiosis for food web dynamics
Journal of Mathematical Biology, 2004
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-nutrienttwo-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.
Modeling the Thermal Behavior of Biosphere 2 in a Non-Controlled Environment Using Bond Graphs
1996
Biosphere 2 is a closed e cological system of high complexity built to deepen the understanding of ecological systems, to study the dynamics of closed e cologies, and to learn to control their behavior. The use of modeling and simulation is crucial in the achievement of these goals. Understanding a physical system is almost synonymous with possessing a mod e l o f i t s comportment. The main goal of this study is the development of a dynamic bond graph model that represents the thermal behavior of the complex ecological system under study, Biosphere 2. In this work, a rst model that captures the behavior of the ecological system in a noncontrolled environment is presented.
Qualitative Analysis of Food-Web Model through Diffusion-Driven Instability
Mathematics and Statistics, 2022
Many food webs exist in the ecosystem, and their survival is directly dependent on the growth rate of primary prey; it balances the entire ecosystem. The spatiotemporal dynamics of three species' food webs was proposed and analyzed in this paper, where the intermediate predator's predation term follows Holling Type IV and the top predator's predation term follows Holling Type II. To begin, we examine the system's stability using linear stability analysis. We first obtained an equilibrium solution set and then used a Jacobian method to investigate the system's stability at a biologically feasible equilibrium point. We investigate random movement in species in the presence of diffusion, establish conditions for system stability, and derive the Turing instability condition. Following that, the Turing instability condition for a spatial food web system is calculated. Finally, numerical simulations are used to validate the findings. We discovered several intriguing spatial patterns (spots, strip, and mixed patterns) that help us understand the dynamics of the real-world food web. As a result, the Turing instability analysis used in the complex food web system is especially relevant experimentally because the associated consequences can be researched and applied to a wide range of mathematical, ecological, and biological models.
Food Web Assembly Rules for Generalized Lotka-Volterra Equations
PLOS Computational Biology, 2016
In food webs, many interacting species coexist despite the restrictions imposed by the competitive exclusion principle and apparent competition. For the generalized Lotka-Volterra equations, sustainable coexistence necessitates nonzero determinant of the interaction matrix. Here we show that this requirement is equivalent to demanding that each species be part of a non-overlapping pairing, which substantially constrains the food web structure. We demonstrate that a stable food web can always be obtained if a non-overlapping pairing exists. If it does not, the matrix rank can be used to quantify the lack of niches, corresponding to unpaired species. For the species richness at each trophic level, we derive the food web assembly rules, which specify sustainable combinations. In neighboring levels, these rules allow the higher level to avert competitive exclusion at the lower, thereby incorporating apparent competition. In agreement with data, the assembly rules predict high species numbers at intermediate levels and thinning at the top and bottom. Using comprehensive food web data, we demonstrate how omnivores or parasites with hosts at multiple trophic levels can loosen the constraints and help obtain coexistence in food webs. Hence, omnivory may be the glue that keeps communities intact even under extinction or ecological release of species.
The Complex Dynamics in a Food Chain Involving Different Functional Responses
Iraqi journal of science, 2022
A food chain model in which the top predator growing logistically has been proposed and studied. Two types of Holling's functional responses type IV and type II have been used in the first trophic level and second trophic level respectively, in addition to Leslie-Gower in the third level. The properties of the solution are discussed. Since the boundary dynamics are affecting the dynamical behavior of the whole dynamical system, the linearization technique is used to study the stability of the subsystem of the proposed model. The persistence conditions of the obtained subsystem of the food chain are established. Finally, the model is simulated numerically to understand the global dynamics of the food chain under study.
A mathematical model of facultative mutualism with populations interacting in a food chain
Mathematical Biosciences, 1989
Facultative mutualism with populations interacting in a food chain is modeled by a system of four autonomous ordinary differential equations. Two cases are considered: mutualism with the prey and mutual&m with the first predator. In both cases persistence and extinction criteria are developed in terms of the invariant flows on the boundaries. *Research is partly based on a doctoral thesis at the University of Alberta. +Research is partly supported by the Natural Sciences and Engineering Research Council of Canada, grant NSERC A4823.
Stability of a model food web
Physical review. E, Statistical, nonlinear, and soft matter physics, 2009
We investigate numerically the stability of a model food web, introduced by Nunes Amaral and Meyer [Phys. Rev. Lett. 82, 652 (1999)]. The model describes a system of species located in niches at several levels. Upper level species are predating on those from a lower level. We show that the model web is more stable when it is larger, although the number of niches is more important than the number of levels. The food web is self-organizing itself, trying to reach a certain degree of complexity, i.e., number of species and links among them. If the system cannot achieve this state, it will go extinct. We demonstrate that the average number of links per species and the reduced number of species depend in the same way on the number of niches. We also determine how the stability of the food web depends on another parameter of the model, the killing probability. Despite keeping the ratio of the creation and killing probabilities constant, increasing the latter reduces significantly the stab...
Related papers
On the dynamical behavior of three species food web model
Chaos Solitons & Fractals, 2007
In this paper, a mathematical model consisting of two preys one predator with Beddington–DeAngelis functional response is proposed and analyzed. The local stability analysis of the system is carried out. The necessary and sufficient conditions for the persistence of three species food web model are obtained. For the biologically reasonable range of parameter values, the global dynamics of the system has been investigated numerically. Number of bifurcation diagrams has been obtained; Lyapunov exponents have been computed for different attractor sets. It is observed that the model has different types of attractors including chaos.
Consequences of population models for the dynamics of food chains
Mathematical Biosciences, 1998
A class of bioenergetic ecological models is studied for the dynamics of food chains with a nutrient at the base. A constant in¯ux rate of the nutrient and a constant eux rate for all trophic levels is assumed. Starting point is a simple model where prey is converted into predator with a ®xed eciency. This model is extended by the introduction of maintenance and energy reserves at all trophic levels, with two state variables for each trophic level, biomass and reserve energy. Then the dynamics of each population are described by two ordinary dierential equations. For all models the bifurcation diagram for the bi-trophic food chain is simple. There are three important regions; a region where the predator goes to extinction, a region where there is a stable equilibrium and a region where a stable limit cycle exists. Bifurcation diagrams for tritrophic food chains are more complicated. Flip bifurcation curves mark regions where complex dynamic behaviour (higher periodic limit cycles as well as chaotic attractors) can occur. We show numerically that Shil'nikov homoclinic orbits to saddle-focus equilibria exists. The codimension 1 continuations of these orbits form a`skeleton' for a cascade of¯ip and tangent bifurcations. The bifurcation analysis facilitates the study of the consequences of the population model for the dynamic behaviour of a food chain. Although the predicted transient dynamics of a food chain may depend sensitively on the underlying model for the populations, the global picture of the bifurcation diagram for the dierent models is about the same. Ó 1998 Elsevier Science Inc. All rights reserved.
Ecological subsystems via graph theory: the role of strongly connected components
Oikos, 2005
2005. Ecological subsystems via graph theory: the role of strongly connected components. Á/ Oikos 110: 164 Á/176.
Analysis of the dynamics of a realistic ecological model
Chaos, Solitons & Fractals, 2002
A fairly realistic three-species food chain model based on the Leslie±Gower scheme is investigated by using tools borrowed from the nonlinear dynamical systems theory. It is observed that two co-existing attractors may be generated by this ecological model. A type-I intermittency is characterized and a homoclinic orbit is found. Ó : S 0 9 6 0 -0 7 7 9 ( 0 0 ) 0 0 2 3 9 -3
Hamiltonian dynamics for complex food webs
Physical Review E, 2016
We investigate stability and dynamics of large ecological networks by introducing classical methods of dynamical system theory from physics, including Hamiltonian and averaging methods. Our analysis exploits the topological structure of the network, namely the existence of strongly connected nodes (hubs) in the networks. We reveal new relations between topology, interaction structure, and network dynamics. We describe mechanisms of catastrophic phenomena leading to sharp changes of dynamics and hence completely altering the ecosystem. We also show how these phenomena depend on the structure of interaction between species. We can conclude that a Hamiltonian structure of biological interactions leads to stability and large biodiversity.
Study of a tri-trophic prey-dependent food chain model of interacting populations
Mathematical Biosciences, 2013
The current paper accounts for the influence of intra-specific competition among predators in a prey dependent tri-trophic food chain model of interacting populations. We offer a detailed mathematical analysis of the proposed food chain model to illustrate some of the significant results that has arisen from the interplay of deterministic ecological phenomena and processes. Biologically feasible equilibria of the system are observed and the behaviours of the system around each of them are described. In particular, persistence, stability (local and global) and bifurcation (saddle-node, transcritical, Hopf-Andronov) analysis of this model are obtained. Relevant results from previous well known food chain models are compared with the current findings. Global stability analysis is also carried out by constructing appropriate Lyapunov functions. Numerical simulations show that the present system is capable enough to produce chaotic dynamics when the rate of self-interaction is very low. On the other hand such chaotic behaviour disappears for a certain value of the rate of self interaction. In addition, numerical simulations with experimented parameters values confirm the analytical results and shows that intra-specific competitions bears a potential role in controlling the chaotic dynamics of the system; and thus the role of self interactions in food chain model is illustrated first time. Finally, a discussion of the ecological applications of the analytical and numerical findings concludes the paper.
Analysis of Dynamics in a Complex Food Chain with Ratio-Dependent Functional Response
In this paper, we study a new model obtained as an extension of a three-species food chain model with ratio-dependent functional response. We provide non-persistence and permanence results and investigate the stability of all possible equilibria in relation to the ecological parameters. Results are obtained for the trivial and prey-only equilibria where the singularity of the model prevents linearization, and the remaining semi-trivial equilibria are studied using linearization. We provide a detailed analysis of conditions for existence, uniqueness, and multiplicity of coexistence equilibria, as well as permanent effect for all species. The complexity of the dynamics in this model is theoretically discussed and graphically demonstrated through various examples and numerical simulations.
Structural food web regimes
Ecological Modelling, 2007
Ecological network analysis allows for an investigation of the structural and functional interconnectedness in ecosystems. Typically, these interactions are seen to comprise a food web of "who eats whom", but more generally applies to the transfer of energy-matter within the biotic and abiotic ecosphere. This web of transactions can be depicted as a digraph or an adjacency matrix in which the presence of direct transactions are represented as a 1 and no transactions as 0. Each transaction between system components leads to an overall network structural pattern. These structures cluster into different categories or regimes based on their cyclic nature. This paper demonstrates threshold effects of the placement or removal of links, such that certain changes essentially keep the structure in the same regime whereas others shift it to another regime in a non-linear manner.
From Small to Large Ecological Networks in a Dynamic World
2005
Food webs are one of the most useful, and challenging, objects of study in ecology. These networks of predator-prey interactions, conjured in Darwin's image of a "tangled bank," provide a paradigmatic example of complex adaptive systems. While it is deceptively easy to throw together simplified caricatures of feeding relationships among a few taxa as can be seen in many basic ecology text books, it is much harder to create detailed descriptions that portray a full range of diversity of species in an ecosystem and the complexity of interactions among them ( ). Difficult to sample, difficult to describe, and difficult to model, food webs are nevertheless of central practical and theoretical importance. The interactions between species on different trophic (feeding) levels underlie the flow of energy and biomass in ecosystems and mediate species' responses to natural and unnatural perturbations such as habitat loss. Understanding the ecology and mathematics of food webs, and more broadly, ecological networks, is central to understanding the fate of biodiversity and ecosystems in response to perturbations.
Eco-bond graphs: an energy-based modeling framework for complex dynamic systems with a focus on sustainability and embodied energy flows
This article presents a general methodology for modeling complex dynamic systems, focusing on sustainability properties that emerge from tracking energy flows. We adopt the embodied energy (emergy) concept that traces all energy transformations required for running a process. Energy can therefore be studied in terms of all energy previously invested up to the primary sources, and sustainability can be analyzed structurally. These ideas were implemented in the bond graph framework, a modeling paradigm where variables are explicitly checked for adherence to energy conservation principles. We introduced the new Ecological Bond Graphs (EcoBG) along with the EcoBondLib Modelica library. EcoBG represent systems in a three-faceted fashion, describing dynamics at their mass, energy, and emergy facets. EcoBG offers a scalable formalism for the description of emergy dynamic equations (resolving some mathematical difficulties present in their original formulation) and new capabilities for detecting unsustainable phases not automatically discovered when using the emergy technique alone.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.