Cats Are Not Fish: A Ricker Model Fails to Account for Key Aspects of Trap–Neuter–Return Programs

In a frequently cited 2005 paper, a Ricker model was used to assess the effectiveness of trap–neuter–return (TNR) programs for managing free-roaming domestic cat populations. The model (which was originally developed for application in the management of fisheries) used data obtained from two countywide programs, and the results indicated that any population reductions, if they existed, were at best modest. In the present study, we applied the same analysis methods to data from two long-term (i.e., >20 years) TNR programs for which significant population reductions have been documented. Our results revealed that the model cannot account for some key aspects of typical TNR programs, and the wild population swings it predicts do not correspond to the relative stability of free-roaming cat populations. A Ricker model is therefore inappropriate for use in assessing the effectiveness of TNR programs. A more recently developed, stochastic model, which accounts for the movement of cats in and out of a given area, is better suited for predicting the sterilization effort necessary to reduce free-roaming cat numbers through TNR programs.

Phase Multistability of Dynamics Modes of the Ricker Model with Periodic Malthusian Parameter

The paper investigates the phase multistability of dynamical modes of the Ricker model with 2-year periodic Malthusian parameter. It is shown that both the variable perturbation and the phase shift of the Malthusian parameter can lead to a phase shift or a change in the dynamic mode observed. The possibility of switches between different dynamic modes is due to multistability, since the model has two different stable 2-cycles. The first stable 2-cycle is the result of transcritical bifurcation and is synchronous to the oscillations of the Malthusian parameter. The second stable 2-cycle arises as a result of the tangent bifurcation and is asynchronous to the oscillations of the Malthusian parameter. This indicates that two-year fluctuations in the population size can be both synchronous and asynchronous to the fluctuations in the environment. The phase shift of the Malthusian parameter causes a phase shift in the stable 4-cycle of the first bifurcation series to one or even three elements of the 4-cycle. The phase shift to two elements of this 4-cycle is possible due to a change in the half-amplitude of the Malthusian parameter oscillation or the variable perturbation. At the same time, the longer period of the cycle, the more phases with their attraction basins it has, and the smaller the threshold values above which shift from the attraction basin to another one occur. As a result, in the case of cycles with long period (for example, 8-cycle) perturbations, that stable cycles with short period are able to "absorb", can cause different phase transitions, which significantly complicates the dynamics of the model trajectory and, as a consequence, the identification of the dynamic mode observed.

Mathematical Modeling of Dynamics of the Number of Specimens in a Biological Population Under Changing External Conditions on the Example of the Burzyan Wild-Hive Honeybee (Apismellifera L., 1758)

The usage of a non-autonomous discrete model (Ricker model) for describing the dynamics of a biological population is considered. It is shown that in case of periodic changes in parameters, the model can be reduced into equivalent autonomous system. The problems of determining the model parameters in a situation where these parameters depend on time are discussed. As an application, the problem of mathematical modeling of the dynamics of the number of families of the natural population of the Burzyan wild-hive honeybee living on the territory of the Republic of Bashkortostan is considered. The results convincingly demonstrate the fact that the dynamics of the Burzyan Wild-Hive Honeybee is significantly influenced by a combination of natural factors. For example the sum of the precipitation in February is particularly significant here (in particular, the increase in precipitation affects the number of bees negatively) and the temperature values in March, April and June.

Density dependence in a seasonal time series of the bamboo mosquito,Tripteroides bambusa(Diptera: Culicidae)

The bamboo mosquito,Tripteroides bambusa(Yamada) (Diptera: Culicidae), is a mosquito species ubiquitous across forested landscapes in Japan. During 2014 we sampled adult mosquitoes from May to November using a sweep net in Nagasaki, Japan. We recorded and managed our field data using Open Data Kit, which eased the overall process of data management before performing their statistical analysis. Here, we analyse the resulting biweekly time series of the bamboo mosquito abundance using time-series statistical techniques. Specifically, we test for density dependence in the population dynamics fitting the Ricker model. Parameter estimates for the Ricker model suggest that the bamboo mosquito is under density dependence regulation and that its population dynamics is stable. Our data also suggest the bamboo mosquito increased its abundance when temperature was more variable at our study site. Further work is warranted to better understand the linkage between the observed density dependence in the adults and the larvae of this mosquito species.

Preventing Noise-Induced Extinction in Discrete Population Models

A problem of the analysis and prevention of noise-induced extinction in nonlinear population models is considered. For the solution of this problem, we suggest a general approach based on the stochastic sensitivity analysis. To prevent the noise-induced extinction, we construct feedback regulators which provide a low stochastic sensitivity and keep the system close to the safe equilibrium regime. For the demonstration of this approach, we apply our mathematical technique to the conceptual but quite representative Ricker-type models. A variant of the Ricker model with delay is studied along with the classic widely used one-dimensional system.

A Mechanistic Stochastic Ricker Model: Analytical and Numerical Investigations

The Ricker model is one of the simplest and most widely-used ecological models displaying complex nonlinear dynamics. We study a discrete-time population model, which is derived from simple assumptions concerning individual organisms’ behavior, using the “site-based” approach, developed by Brännström, Broomhead, Johansson and Sumpter. In the large-population limit the model converges to the Ricker model, and can thus be considered a mechanistic version of the Ricker model, derived from basic ecological principles, and taking into account the demographic stochasticity inherent to finite populations. We employ several analytical and precise numerical methods to study the model, showing how each approach contributes to understanding the model’s dynamics. Expressing the model as a Markov chain, we employ the concept of quasi-stationary distributions, which are computed numerically, and used to examine the interaction between complex deterministic dynamics and demographic stochasticity, as well as to calculate mean times to extinction. A Gaussian Markov chain approximation is used to obtain quantitative asymptotic approximations for the size of fluctuations of the stochastic model’s time series around the deterministic trajectory, and for the correlations between successive fluctuations. Results of these approximations are compared to results obtained from quasi-stationary distributions and from direct simulations, and are shown to be in good agreement.