Physiologically Structured Population Models: Towards a General Mathematical Theory

Diekmann, Odo; Gyllenberg, Mats; Metz, J.A.J.

doi:10.1007/978-3-540-34428-5_2

Cited by 19 publications

(21 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, our results are directly related to and in support of the work of Diekmann et al [5] toward a general mathematical theory for physiologically structured population models (see [6,7]). …”

Section: Resultssupporting

confidence: 85%

“…In this situation the model can be reduced to the age-structured case. Diekmann et al [5][6][7] have been working on the Principle of Linearized Stability for general, physiologically structured population models. Their work-once completed successfully-would rigorously link the linear and nonlinear stability of solutions for the model studied here.…”

Section: Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Stability and regularity results for a size-structured population model

Farkas

Hagen

2007

Journal of Mathematical Analysis and Applications

116

View full text Add to dashboard Cite

In the present paper a nonlinear size-structured population dynamical model with size and density dependent vital rate functions is considered. The linearization about stationary solutions is analyzed by semigroup and spectral methods. In particular, the spectrally determined growth property of the linearized semigroup is derived from its long-term regularity. These analytical results make it possible to derive linear stability and instability results under biologically meaningful conditions on the vital rates. The principal stability criteria are given in terms of a modified net reproduction rate.

show abstract

Section: Resultssupporting

confidence: 85%

Section: Resultsmentioning

confidence: 99%

Stability and regularity results for a size-structured population model

Farkas

Hagen

2007

Journal of Mathematical Analysis and Applications

116

View full text Add to dashboard Cite

show abstract

“…Reasoning from general assumptions, rather than specific models, is a relatively new and potentially powerful approach in evolutionary theory. This approach has previously been applied to the dynamics of physiologically structured populations (Metz and Diekmann, 1986;Diekmann et al, 1998Diekmann et al, , 2001Diekmann et al, , 2007Durinx et al, 2008), evolutionary game theory (Tarnita et al, 2009b(Tarnita et al, , 2011, quantitative trait evolution (Champagnat et al, 2006;Simon, 2008), and models with fixed spatial structure . The advantage of this approach is that general statements, applying to many models or systems, can be proven all at once.…”

Section: General Approaches To Evolutionary Theorymentioning

confidence: 99%

Adaptive Dynamics with Interaction Structure

Allen

Nowak

Dieckmann

2013

The American Naturalist

View full text Add to dashboard Cite

Evolutionary dynamics depend critically on a population's interaction structure-the pattern of which individuals interact with which others, depending on the state of the population and the environment. Previous research has shown, for example, that cooperative behaviors disfavored in well-mixed populations can be favored when interactions occur only between spatial neighbors or group members. Combining the adaptive dynamics approach with recent advances in evolutionary game theory, we here introduce a general mathematical framework for analyzing the long-term evolution of continuous game strategies for a broad class of evolutionary models, encompassing many varieties of interaction structure. Our main result, the "canonical equation of adaptive dynamics with interaction structure", characterizes expected evolutionary trajectories resulting from any such model, thereby generalizing a central tool of adaptive dynamics theory. Interestingly, the effects of different interaction structures and update rules on evolutionary trajectories are fully captured by just two real numbers associated with each model, which are independent of the considered game. The first, a structure coefficient, quantifies the effects on selection pressures, and thus on the shapes of expected evolutionary trajectories. The second, an effective population size, quantifies the effects on selection responses, and thus on the expected rates of adaptation. Applying our results to two social dilemmas, we show how the range of evolutionarily stable cooperative behaviors systematically varies with a model's structure coefficient.2

show abstract

“…Each of the structured population models discussed in [9,10,11,12,13] (see the well-known monographs on structured population dynamics [5,7,16,19,22] and the more recent related papers [3,4,6], as well for further reference) describe the dynamics of a population living in a closed territory. Thus, the density of newborns (zero or minimal size individuals in the case of size-structured models) is determined by the fertility rate of sexually mature individuals.…”

Section: Introductionmentioning

confidence: 99%

Structured populations: The stabilizing effect of the inflow of newborns from an external source and the net growth rate

Farkas

2008

Applied Mathematics and Computation

View full text Add to dashboard Cite

We investigate the effect of a positive population inflow of individuals from an external source on the dynamical behaviour of certain physisologically structured population models. We treat a size-structured model with constant inflow and nonlinear birth rate and an age-structured model with nonlinear (density dependent) inflow and linear birth rate. Analogously to the inherent net reproduction rate we introduce a net growth rate and discuss how this net growth rate can be related to our stability/instability conditions.

show abstract

Physiologically Structured Population Models: Towards a General Mathematical Theory

Cited by 19 publications

References 21 publications

Stability and regularity results for a size-structured population model

Stability and regularity results for a size-structured population model

Adaptive Dynamics with Interaction Structure

Structured populations: The stabilizing effect of the inflow of newborns from an external source and the net growth rate

Contact Info

Product

Resources

About