Global stability of an age-structured population model on several temporally variable patches

We consider an age-structured density-dependent population model on several temporally variable patches. There are two key assumptions on which we base model setup and analysis. First, intraspecific competition is limited to competition between individuals of the same age (pure intra-cohort competition) and it affects density-dependent mortality. Second, dispersal between patches ensures that each patch can be reached from every other patch, directly or through several intermediary patches, within individual reproductive age. Using strong monotonicity we prove existence and uniqueness of solution and analyze its large-time behavior in cases of constant, periodically variable and irregularly variable environment. In analogy to the next generation operator, we introduce the net reproductive operator and the basic reproduction number $$R_0$$ R 0 for time-independent and periodical models and establish the permanence dichotomy: if $$R_0\le 1$$ R 0 ≤ 1 , extinction on all patches is imminent, and if $$R_0>1$$ R 0 > 1 , permanence on all patches is guaranteed. We show that a solution for the general time-dependent problem can be bounded by above and below by solutions to the associated periodic problems. Using two-side estimates, we establish uniform boundedness and uniform persistence of a solution for the general time-dependent problem and describe its asymptotic behaviour.

Dynamics of a new stage-structured population model with transient and nontransient impulsive effects in a polluted environment

In this paper, we consider a new stage-structured population model with transient and nontransient impulsive effects in a polluted environment. By using the theories of impulsive differential equations, we obtain the globally asymptotically stable condition of a population-extinction solution; we also present the permanent condition for the investigated system. The results indicate that the nontransient and transient impulsive harvesting rate play important roles in system permanence. Finally, numerical analyses are carried out to illustrate the results. Our results provide effective methods for biological resource management in a polluted environment.

An Application of the Z-Transform in Extracting the Density Regulation from the Periodic Output of an Age-Structured Population Model

The application of the Z-transform, a manipulation tool from the discrete signal processing (DSP) toolbox, on an ecological model was motivated by the mathematical similarities between an age-structured fish population model with a non linear density regulation and a linear time invariant (LTI) control system. Both models include a switching mechanism in regulating stock/signal throughput in accordance with a given density limitation/set value and both models can be expressed in terms of a negative feedback loop difference equations (Getz & Haight,1989; °Astr¨om & Murray, 2008). In the fish model, the switching mechanism is a density regulated stock-recruitment (SR) function which models the strategies implemented by the population in keeping the vulnerable egg-larvaejuvenile densities within an environmental limitation thereof (Subbey et al, 2014). A switching mechanism is also present in control engineering, for example, in the mechanism associated with cruise control in cars which keeps traveling speed close to a chosen set value midst varying weather and road conditions (Antsaklis and Gao, 2005). In both cases, the choosing of the control action and the tuning of its parameters requires careful consideration to avoid failures such as incorrectly timed switching actions in a control plant (see Kuphaldt (2019)) and errors in estimating total allowable catch (TAC) in the fishing industry (see Borlestean et al (2015), Skagen et al (2013) and Taboadai and R. Anadn (2016)). The Z-transform has proven itself useful in tuning LTI controlmodels for a desired control action (see Orfanidis, (2010) and Smith, (1999)) and it is on this account that its application was extended to the ecological model in pursuit of a more efficient way of estimating SR parameters to simulate an already existing output. It was however found that it could not be used for parameter tuning but rather for the extraction of the SR component hidden in the output together with components resulting from the age structure itself. Such an extraction can greatly assist in the mathematical identification of the SR, reducing the complexity of its choosing as there are many different types used in the fishing industry such as the classic Beverton-Holt model, the Ricker model and Shepherd model (Myers, 2001; Iles, 1994; Shepherd, 1982). It can also be used to monitor changes in the SR over time which can indicate the presence of strategy evolution (Apaloo et al, 2009; Br¨annstr¨om et al, 2013). In 1998 Schoombie and Getz investigated the latter by subjecting the Shepherd SR to strategy optimization with regards to a parameter associated with population interventions in regulating recruitment throughput and it is because of this versatility that the Shepherd SR is chosen for the intended extraction. In true control style, Simulink, a graphic environment for designing control simulations, is used to visualize the production of the output as well as the extraction of the SR from it. This paper showcases the versatility of the Z transform and the possibilities and unexpected finds when applied to similar systems designed to regulate signals or, in this case, recruitment densities.

Evolution of reproductive strategies in incipient multicellularity

Multicellular organisms can potentially show a large degree of diversity in reproductive strategies, as they could reproduce offspring with varying sizes and compositions compared to their unicellular ancestors. In reality, only a few of these reproductive strategies are prevalent. To understand why this could be the case, we develop a stage-structured population model to probe the evolutionary growth advantages of reproductive strategies in incipient multicellular organisms. The performance of reproductive strategies is evaluated by the growth rates of corresponding populations. We identify the optimal reproductive strategy, which leads to the largest growth rate for a population. Considering the effects of organism size and cellular interaction, we found that distinct reproductive strategies could perform uniquely or equally well under different conditions. Only binary-splitting reproductive strategies can be uniquely optimal. Our results show that organism size and cellular interaction can play crucial roles in shaping reproductive strategies in nascent multicellularity. Our model sheds light on understanding the mechanism driving the evolution of reproductive strategies in incipient multicellularity. Meanwhile, beyond multicellularity, our results imply a crucial factor in the evolution of reproductive strategies of unicellular species - organism size.

Evolution of heterogeneity under constant and variable environments

Various definitions of fitness are essentially based on the number of descendants of an allele or a phenotype after a sufficiently long time. However, these different definitions do not explicate the continuous evolution of life histories. Herein, we focus on the eigenfunction of an age-structured population model as fitness. The function generates an equation, called the Hamilton–Jacobi–Bellman equation, that achieves adaptive control of life history in terms of both the presence and absence of the density effect. Further, we introduce a perturbation method that applies the solution of this equation to the long-term logarithmic growth rate of a stochastic structured population model. We adopt this method to realize the adaptive control of heterogeneity for an optimal foraging problem in a variable environment as the analyzable example. The result indicates that the eigenfunction is involved in adaptive strategies under all the environments listed herein. Thus, we aim to systematize adaptive life histories in the presence of density effects and variable environments using the proposed objective function as a universal fitness candidate.

The regulating effect of growth plasticity on the dynamics of structured populations

Plasticity is the extent to which life history processes such as growth and reproduction depend on the environment. Plasticity in individual growth varies widely between taxa. Nonetheless, little is known about the effect of plasticity in individual growth on the ecological dynamics of populations. In this article we analyse a physiologically structured population model of a consumer population in which the individual growth rate can be varied between entirely plastic to entirely non-plastic. We derive this population level model from a dynamic energy budget model to ensure an accurate energetic coupling between ingestion, somatic maintenance, growth, and reproduction within an individual. We show that the consumer population is either limited by adult fecundity or juvenile survival up to maturation, depending on the level of growth plasticity and the non-plastic individual growth rate. Under these two regimes we also find two different types of population cycles which again arise due to fluctuation in respectively juvenile growth rate or adult fecundity. In the end our model not only provides insight into the effects of growth plasticity on population dynamics, but also provides a link between the dynamics found in age- and size-structured models.

Competition in depleting resource environments shapes the thermal response of population fitness in a disease vector

Mathematical models that incorporate the temperature dependence of lab-measured life history traits are increasingly used to predict how climatic warming will affect ectotherms, including disease vectors and other arthropods. These temperature-trait relationships are typically measured under laboratory conditions that ignore how conspecific competition in depleting resource environments—a commonly occurring scenario in nature—regulates natural populations. Here, we used laboratory experiments on the mosquito Aedes aegypti, combined with a stage-structured population model, to show that intensified larval competition in ecologically-realistic depleting resource environments can significantly diminish the vector’s maximal population-level fitness across the entire temperature range, cause a 6°C decrease in the optimal temperature for fitness, and contract its thermal niche width by 10°C. Our results provide evidence for future studies to consider competition dynamics under depleting resources when predicting how eukaryotic ectotherms will respond to climatic warming.

Symbiosis increases population size and buffers environmental fluctuations in a physiologically-structured model parameterized for thyasirid bivalves

Symbioses whereby one partner provisions a nutritional resource to the other may alter energy allocation towards reproduction and survival in the recipient partner, potentially impacting population dynamics. Asymbiotic thyasirid bivalves feed predominantly on free-living bacteria, which fluctuate in abundance due to seasonality-driven temperature variations. Symbiotic thyasirids are mixotrophs, gaining nutrients from free-living bacteria and symbiotic bacteria that they host on their enlarged gills. Symbiotic bacteria may function as an additional energy reserve for thyasirids, allowing the hosts to allocate more energy to reproduction. We hypothesize that, for symbiotic thyasirids, the symbionts are a nutritional source that mitigates resource limitation. Using Dynamic Energy Budget theory, we built a physiologically-structured population model assuming equal mortality rates in both species. We find that without seasonal fluctuations, symbiotic thyasirids have higher abundances than asymbiotic thyasirids since the symbionts increase reproduction. Both species have similar population sizes in fluctuating environments, suggesting different adaptations to seasonality: asymbiotic thyasirids have adapted their physiology, while symbiotic thyasirids have adapted through mixotrophy. Our results highlight the significance of linking individual energetics and life-history traits to population dynamics and are the first step towards understanding the role of symbioses in population and community dynamics.