Alfonso Ruiz-Herrera

The hydra effect, bubbles, and chaos in a simple discrete population model with constant effort harvesting

We analyze the effects of a strategy of constant effort harvesting in the global dynamics of a one-dimensional discrete population model that includes density-independent survivorship of adults and overcompensating density dependence. We discuss the phenomenon of bubbling (which indicates that harvesting can magnify fluctuations in population abundance) and the hydra effect, which means that the stock size gets larger as harvesting rate increases. Moreover, we show that the system displays chaotic behaviour under the combination of high per capita recruitment and small survivorship rates.

To connect or not to connect isolated patches

Empirical evidence suggests that dispersal can have different effects on the time evolution of a spatially structured population. In this study, we explored the impact of the migration rate in a coupled map lattice system. To capture this impact, we assumed symmetric dispersal and simple dynamics in the local populations. However, we allowed heterogeneity between the patches, including both source-source and source-sink systems. Our results show that this simple theoretical setting has the potential to unify the diversity of behaviours of the total population size observed in previous studies. Indeed, we found that the response of the total population size to migration was non-monotone in source-source and many source-sink situations, thereby suggesting that an increase in the dispersal rate could be related to either an increase or a decrease in the total population size. As we will illustrate, this response provides a possible theoretical explanation of some benefits of control strategies involving spatial considerations as no-take zones. Our study also analyses the impact of the migration rate on persistence, spatial coherence, and initial transients. This was motivated by previous theoretical observations in coupled systems that the rate of migration affects these three aspects. Related to persistence, we rigorously extended a previous result from the linear to the non-linear case. This result essentially states that persistence depends on the stability of the origin. On the other hand, we stress that negative effects due to an increase of the spatial coherence could be neutralised by the unimodal response of the total population size. Finally, the study of the initial transients, which is relevant for interpreting experimental results, highlights that the relationship between the rate of migration and the total population size remains the same even in the transient phase.

Delayed population models with Allee effects and exploitation.

Allee effects make populations more vulnerable to extinction, especially under severe harvesting or predation. Using a delay-differential equation modeling the evolution of a single-species population subject to constant effort harvesting, we show that the interplay between harvest strength and Allee effects leads not only to collapses due to overexploitation; large delays can interact with Allee effects to produce extinction at population densities that would survive for smaller time delays. In case of bistability, our estimations on the basins of attraction of the two coexisting attractors improve some recent results in this direction. Moreover, we show that the persistent attractor can exhibit bubbling: a stable equilibrium loses its stability as harvesting effort increases, giving rise to sustained oscillations, but higher mortality rates stabilize the equilibrium again.

Exclusion and dominance in discrete population models via the carrying simplex

This paper is devoted to show that Hirschs results on the existence of a carrying simplex are a powerful tool to understand the dynamics of Kolmogorov models. For two and three species, we prove that there is exclusion for our models if and only if there are no coexistence states. The proof of this result is based on a result in planar topology due to Campos, Ortega and Tineo. For an arbitrary number of species, we will obtain dominance criteria following the notions of Franke and Yakubu. In this scenario, the crucial fact will be that the carrying simplex is an unordered manifold. Applications in concrete models are given.

Chaos in Discrete Structured Population Models

We prove analytically the existence of chaotic dynamics in some classical discrete-time age-structured population models. Our approach allows us to estimate the sensitive dependence on the initial conditions, regions of initial data with chaotic behavior, and explicit ranges of parameters for which the considered models display chaos. These properties have important implications for evaluating the influence of a chaotic regime on the predictions based on mathematical models. We illustrate through particular examples how to apply our results.

Some examples related to the method of Lagrangian descriptors.

We provide families of counter-examples, including Hamiltonian systems, to the method of Lagrangian descriptors developed by Mancho, Wiggins, and their co-workers. A detailed mathematical discussion on why that methodology fails together with some pathological phenomena are given as well.

Periodic Solutions and Chaotic Dynamics in Forced Impact Oscillators

It is shown that a periodically forced impact oscillator may exhibit chaotic dynamics on two symbols, as well as an infinity of periodic solutions. Two cases are considered, depending on if the impact velocity is finite or infinite. In the second case, the Poincare map is well defined by continuation of the energy. The proof combines the study of phase-plane curves together with the “stretching-along-paths” notion.

Potential Impact of Carry-Over Effects in the Dynamics and Management of Seasonal Populations

For many species living in changing environments, processes during one season influence vital rates in a subsequent season in the same annual cycle. The interplay between these carry-over effects between seasons and other density-dependent events can have a strong influence on population size and variability. We carry out a theoretical study of a discrete semelparous population model with an annual cycle divided into a breeding and a non-breeding season; the model assumes carry-over effects coming from the non-breeding period and affecting breeding performance through a density-dependent adjustment of the growth rate parameter. We analyze the influence of carry-over effects on population size, focusing on two important aspects: compensatory mortality and population variability. To understand the potential consequences of carry-over effects for management, we have introduced constant effort harvesting in the model. Our results show that carry-over effects may induce dramatic changes in population stability as harvesting pressure is increased, but these changes strongly depend on whether harvesting occurs prior to reproduction or after it.

Analysis of dispersal effects in metapopulation models

The interplay between local dynamics and dispersal rates in discrete metapopulation models for homogeneous landscapes is studied. We introduce an approach based on scalar dynamics to study global attraction of equilibria and periodic orbits. This approach applies for any number of patches, dispersal rates, or landscape structure. The existence of chaos in metapopulation models is also discussed. We analyze issues such as sensitive dependence on the initial conditions or short/intermediate/long term behaviours of chaotic orbits.

Topological Criteria of Global Attraction with Applications in Population Dynamics

In this paper we derive a criterion of trivial dynamics based on the theory of translation arcs. This criterion extends and unifies some results in the literature. Applications in continuous and discrete models of population dynamics are given.