Mathematical Models in Biology
01 Feb 2005-
TL;DR: The theory of linear difference equations applied to population growth and the applications of nonlinear difference equations to population biology are explained.
Abstract: Part I. Discrete Process in Biology: 1. The theory of linear difference equations applied to population growth 2. Nonlinear difference equations 3. Applications of nonlinear difference equations to population biology Part II. Continuous Processes and Ordinary Differential Equations: 4. An introduction to continuous models 5. Phase-plane methods and qualitative solutions 6. Applications of continuous models to population dynamics 7. Models for molecular events 8. Limit cycles, oscillations, and excitable systems Part III. Spatially Distributed Systems and Partial Differential Equation Models: 9. An introduction to partial differential equations and diffusion in biological settings 10. Partial differential equation models in biology 11. Models for development and pattern formation in biological systems Selected answers Author index Subject index.
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TL;DR: The construction of a genetic toggle switch is presented—a synthetic, bistable gene-regulatory network—in Escherichia coli and a simple theory is provided that predicts the conditions necessary for bistability.
Abstract: It has been proposed' that gene-regulatory circuits with virtually any desired property can be constructed from networks of simple regulatory elements. These properties, which include multistability and oscillations, have been found in specialized gene circuits such as the bacteriophage lambda switch and the Cyanobacteria circadian oscillator. However, these behaviours have not been demonstrated in networks of non-specialized regulatory components. Here we present the construction of a genetic toggle switch-a synthetic, bistable gene-regulatory network-in Escherichia coli and provide a simple theory that predicts the conditions necessary for bistability. The toggle is constructed from any two repressible promoters arranged in a mutually inhibitory network. It is flipped between stable states using transient chemical or thermal induction and exhibits a nearly ideal switching threshold. As a practical device, the toggle switch forms a synthetic, addressable cellular memory unit and has implications for biotechnology, biocomputing and gene therapy.
4,222 citations
TL;DR: This model simultaneously explains the coexistence of behavioural types, the consistency of behaviour through time and the structure of behavioural correlations across contexts, and explains the common finding that explorative behaviour and risk-related traits like boldness and aggressiveness are common characteristics of animal personalities.
Abstract: In recent years evidence has been accumulating that personalities are not only found in humans but also in a wide range of other animal species. Individuals differ consistently in their behavioural tendencies and the behaviour in one context is correlated with the behaviour in multiple other contexts. From an adaptive perspective, the evolution of animal personalities is still a mystery, because a more flexible structure of behaviour should provide a selective advantage. Accordingly, many researchers view personalities as resulting from constraints imposed by the architecture of behaviour (but see ref. 12). In contrast, we show here that animal personalities can be given an adaptive explanation. Our argument is based on the insight that the trade-off between current and future reproduction often results in polymorphic populations in which some individuals put more emphasis on future fitness returns than others. Life-history theory predicts that such differences in fitness expectations should result in systematic differences in risk-taking behaviour. Individuals with high future expectations (who have much to lose) should be more risk-averse than individuals with low expectations. This applies to all kinds of risky situations, so individuals should consistently differ in their behaviour. By means of an evolutionary model we demonstrate that this basic principle results in the evolution of animal personalities. It simultaneously explains the coexistence of behavioural types, the consistency of behaviour through time and the structure of behavioural correlations across contexts. Moreover, it explains the common finding that explorative behaviour and risk-related traits like boldness and aggressiveness are common characteristics of animal personalities.
1,270 citations
TL;DR: In this article, the authors study the foraging behavior of the wandering albatross Diomedea exulans and find a power-law distribution of flight-time intervals.
Abstract: LeVY flights are a special class of random walks whose step lengths are not constant but rather are chosen from a probability distribution with a power-law tail. Realizations of Levy flights in physical phenomena are very diverse, examples including fluid dynamics, dynamical systems, and micelles1,2. This diversity raises the possibility that Levy flights may be found in biological systems. A decade ago, it was proposed that Levy flights may be observed in the behaviour of foraging ants3. Recently, it was argued that Drosophila might perform Levy flights4, but the hypothesis that foraging animals in natural environments perform Levy flights has not been tested. Here we study the foraging behaviour of the wandering albatross Diomedea exulans, and find a power-law distribution of flight-time intervals. We interpret our finding of temporal scale invariance in terms of a scale-invariant spatial distribution of food on the ocean surface. Finally, we examine the significance of our finding in relation to the basis of scale-invariant phenomena observed in biological systems.
1,256 citations
TL;DR: Using a simple experimental protocol, substantial negative feedback on plant growth is found through the soil community, suggesting that it may be involved in the maintenance of plant species diversity.
Abstract: 1 Although its importance for plant mineral nutrition and nutrient cycling has long been recognized, the soil community has rarely been integrated into dynamical frameworks of plant populations, in spite of abundant evidence for its involvement. The concept of feedback may provide theoretical and experimental tools for investigating the importance of the soil community in the population ecology and evolution of plants. 2 A mathematical model demonstrates the potential for two divergent dynamics, with positive feedback leading to the loss of diversity at a local scale and negative feedback leading to its maintenance. A linear contrast of the growth of plants in association with their own soil communities compared to the growth of plants in association with each others' soil communities can be used to differentiate between these possibilities in empirical studies. 3 Spatially explicit computer simulations demonstrate that the dynamics of a spatially structured community, as the soil community is likely to be, can differ from those predicted for a well-mixed population. Specifically, diversity can be maintained between locally homogeneous patches when positive feedback and dispersal occur at local scales. 4 Using a simple experimental protocol, we have found substantial negative feedback on plant growth through the soil community, suggesting that it may be involved in the maintenance of plant species diversity. 5 We discuss the importance of the soil community in other areas of plant ecology and evolution, including the suggestion that interactions with the soil community may be involved in the maintenance of sexual or asexual reproductive systems.
1,029 citations
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13 Feb 2003
794 citations