# Using a (slightly) more realistic model resolves the perfect mixing paradox

## TL;DR

This paper demonstrates that the traditional logistic growth model with r-K parameters effectively describes population dynamics in multi-patch spatial models, resolving the so-called 'perfect mixing paradox' by using more realistic migration assumptions.

## Contribution

The study shows that the classic logistic model remains valid in multi-patch scenarios when appropriate migration functions are used, challenging previous claims that it leads to paradoxical behaviors.

## Key findings

- Traditional logistic model is suitable for multi-patch populations with passive migration.
- The 'perfect mixing paradox' depends on the migration function used in models.
- Biologically realistic migration models do not produce the paradoxical behaviors.

## Abstract

In a recent paper published in Ecosphere, their authors suggest that extending the logistic growth model in its usual r - K parametrization to a multi-patch environment results in undesirable properties, that were referred to as the "perfect mixing paradox". This led the authors to recommend using the Verhulst r - {\alpha} parametrization of the logistic model instead and abandoning the term "carrying capacity" in the context of population dynamics. In this study we show that the use of the logistic equation in its traditional r - K parametrization is appropriate for representing the dynamics of populations in a simple spatial context with passive migration among patches. Furthermore, we show that the conclusions of the mentioned paper depend on the specific functional form for migration rates used for their analyses. In addition, we suggest that their specific migration model was not the best choice since biologically realistic options exist. Moreover, these alternatives require the same number of parameters. The model we present here is free of the paradoxical behaviors presented previously. This allows us to conclude that the logistic growth in its usual r - K parametrization is useful in a spatial context if extended appropriately to a multi-patch scenario and consequently there are no reasons to abandon the concept of carrying capacity. Ecologists should pay great attention when using models in scenarios that are more complex or just different from the ones for which the models were created.