Matrix projection models occupy a central role in population and conservation biology. Matrix models divide a population into discrete classes, even if the structuring trait exhibits continuous variation (e.g., body size). The integral projection model (IPM) avoids discrete classes and potential artifacts from arbitrary class divisions, facilitates parsimonious modeling based on smooth relationships between individual state and demographic performance, and can be implemented with standard matrix software. Here, we extend the IPM to species with complex demographic attributes, including dormant and active life stages, cross‐classification by several attributes (e.g., size, age, and condition), and changes between discrete and continuous structure over the life cycle. We present a general model encompassing these cases, numerical methods, and theoretical results, including stable population growth and sensitivity/elasticity analysis for density‐independent models, local stability analysis in densi...

/pdf/integral-projection-models-for-species-with-complex-2gwelj0ftm.pdf

Integral projection models for species with complex demography.

Part of the Pitman Research Notes in Mathematics series, this text covers: linear evolution equations of parabolic type; semilinear evolution equations of parabolic type; evolution equations and positivity; semilinear periodic evolution equations; and applications.

Abstract Evolution Equations, Periodic Problems and Applications

https://hal.archives-ouvertes.fr/hal-01291211/document

Approximation of the basic reproduction number R0 for vector-borne diseases with a periodic vector population.

We create an analytical structure that reveals the long run risk-return relationship for nonlinear continuous time Markov environments. We do so by studying an eigenvalue problem associated with a positive eigenfunction for a conveniently chosen family of valuation operators. This family forms a semigroup whose members are indexed by the elapsed time between payoff and valuation dates. We represent the semigroup using a positive process with three components: an exponential term constructed from the eigenvalue, a martingale and a transient eigenfunction term. The eigenvalue encodes the risk adjustment, the martingale alters the probability measure to capture long run approximation, and the eigenfunction gives the long run dependence on the Markov state. We establish existence and uniqueness of the relevant eigenvalue and eigenfunction. By showing how changes in the stochastic growth components of cash flows induce changes in the corresponding eigenvalues and eigenfunctions, we reveal a long-run risk return tradeoff.

Long Term Risk: An Operator Approach

/pdf/integral-projection-models-for-species-with-complex-x09nio9pcg.pdf

Integral Projection Models for Species with Complex Demography

Positive linear systems : the method of positive operators

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Positive linear systems : the method of positive operators