# An eco-evolutionary system with naturally bounded traits

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## Abstract

We consider eco-evolutionary processes that include natural bounds on adaptive trait distributions. We implement ecological axioms, that a population grows if it is replete with resources, and doesn’t grow if it has none. These axioms produce natural bounds on the trait means that suggest that the assumption of gamma-distributed traits, where the trait variance is a function of the trait mean, is more appropriate than the usual assumption of normally distributed traits, where the trait variance is independent of the trait mean. We use a Lotka-Volterra model to simulate two plant populations, whose trait means evolve according to an evolutionary model, to simulate populations adapting during invasions. The results of our model simulations using gamma-distributed traits suggest that adapting populations may endure bottlenecks by increasing their fitness and recovering from near extinction to stably coexist. The inclusion of eco-evolutionary processes into ecosystem models generates *K*^{∗} theory which predicts the long-term states of populations that make it through evolutionary bottlenecks. Otherwise, the final states of populations that do not make it through bottlenecks may be predicted by the non-evolutionary *R*^{∗} theory.

## Keywords

Eco-evolutionary modelling*K*

^{∗}theory Gamma-distribution Trait variance Naturally bounded traits Coexistence Competition Invasion

## Notes

### Acknowledgements

The authors thank two anonymous reviewers for their thoughtful and helpful comments that considerably improved this manuscript.

### Author Contributions

Both authors contributed to the conception and design of the study, and the writing of the manuscript. RC did the numerical analysis and produced the figures and data.

### Funding

RC thanks the Mathematical Institute of the University of Oxford for providing funding to support this research.

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