arXiv subjects

Gail S. K. Wolkowicz

Publications and source records attributed to Gail S. K. Wolkowicz.

12 recordsLinked to original sources

arXiv2026

Phase Plane Analysis on Time Scales for a Lotka-Volterra Competition Model

In this work, we formulate a two-species Lotka--Volterra competition model on time scales. To derive the global dynamics of solutions of this planar model, we introduce a dynamic augmented phase plane analysis that extends the traditional phase plane analysis known to be a powerful tool in the analysis of planar differential equations. In the case of a constant graininess, the method is consistent with the augmented phase portrait introduced for the discrete space. However, for non-constant graininess, this augmented phase plane is time-dependent and therefore dynamic. Despite the dynamic character, we are able to identify time-independent information that allows the conclusion of the global dynamics of the introduced Lotka--Volterra competition model on time scales. The dynamic phase plane method, albeit only introduced in the context of this particular two-species competition model, it can be extended to other planar systems on time scales, providing a novel technique to study the global dynamics of planar systems on time scales.

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arXiv2025

Adding a fecundity-survival trade-off to a discrete population model with maturation delay

Although maturation delays are frequently included in population models, researchers rarely account for mortality between birth and maturity. Previous discrete population models have included mortality of immature individuals during the maturation delay finding that increasing the delay decreases the equilibrium population size, eventually leading to extinction. Since maturation delays beyond one breeding cycle are often found in nature, they must also have a benefit leading to a trade-off. We derive a class of models to explore the trade-off between the benefit of a longer maturation delay on fecundity due to larger body sizes at maturity and the down-side on survival. We examine two scenarios: density independent survival and cohort density dependent survival of immature individuals. For the mature and immature individuals, we consider two different, but popular, survival functions: the Beverton--Holt model and the Ricker model. Across all models, we identify a positive maturation delay that maximizes the population size that we refer to as the ``optimal maturation delay'' and a critical delay threshold that results in extinction. We also find oscillatory dynamics with the Ricker survival function for certain ranges of maturation delay. Overall, our delay model sets up a useful phenomenological framework to test multiple combinations of trade-offs in parent survival, offspring survival, and reproductive investment.

q-bio.PE↗

arXiv2024

Competition in the nutrient-driven self-cycling fermentation process

Self-cycling fermentation is an automated process used for culturing microorganisms. We consider a model of $n$ distinct species competing for a single non-reproducing nutrient in a self-cycling fermentor in which the nutrient level is used as the decanting condition. The model is formulated in terms of impulsive ordinary differential equations. We prove that two species are able to coexist in the fermentor under certain conditions. We also provide numerical simulations that suggest coexistence of three species is possible and that competitor-mediated coexistence can occur in this case. These results are in contrast to the chemostat, the continuous analogue, where multiple species cannot coexist on a single nonreproducing nutrient.

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arXiv2022

An augmented phase plane approach for discrete planar maps: Introducing next-iterate operators

The next-iterate operators and corresponding next-iterate root-sets and root-curves associated with the nullclines of a planar discrete map are introduced. How to augment standard phase portraits that include the nullclines and the direction field, by including the signs of the root-operators associated with their nullclines, thus producing an augmented phase portrait, is described. The sign of a next-iterate operator associated with a nullcline determines whether a point is mapped above or below the corresponding nullcline and can, for example, identify positively invariant regions. Using a Lotka-Volterra type competition model, we demonstrate how to construct the augmented phase portrait. We show that the augmented phase portrait provides an elementary, alternative approach for determining the complete global dynamics of this model. We further explore the limitations and potential of the augmented phase portrait by considering a Ricker competition model, a model involving mutualism, and a predator-prey model.

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arXiv2021

Population Growth and Competition Models with Decay and Competition Consistent Delay

We derive an alternative expression for a delayed logistic equation in which the rate of change in the population involves a growth rate that depends on the population density during an earlier time period. In our formulation, the delay in the growth term is consistent with the rate of instantaneous decline in the population given by the model. Our formulation is a modification of [Arino et al., J.~Theoret.~Biol.~241(1):109--119, 2006] by taking the intraspecific competition between the adults and juveniles into account. We provide a complete global analysis showing that no sustained oscillations are possible. A threshold giving the interface between extinction and survival is determined in terms of the parameters in the model. The theory of chain transitive sets and the comparison theorem for cooperative delay differential equations are used to determine the global dynamics of the model. We extend our delayed logistic equation to a system modeling the competition of two species. For the competition model, we provide results on local stability, bifurcation diagrams, and adaptive dynamics. Assuming that the species with shorter delay produces fewer offspring at a time than the species with longer delay, we show that there is a critical value, $τ^*$, such that the evolutionary trend is for the delay to approach $τ^*$.

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arXiv2020

An alternative delayed population growth difference equation model

We propose an alternative delayed population growth difference equation model based on a modification of the Beverton-Holt recurrence, assuming a delay only in the growth contribution that takes into account that those individuals that die during the delay, do not contribute to growth. The model introduced differs from existing delay difference equations in population dynamics, such as the delayed logistic difference equation, which was formulated as a discretization of the Hutchinson model. The analysis of our delayed difference equation model identifies an important critical delay threshold. If the time delay exceeds this threshold, the model predicts that the population will go extinct for all non-negative initial conditions and if it is below this threshold, the population survives and its size converges to a positive globally asymptotically stable equilibrium that is decreasing in size as the delay increases. Firstly, we obtain the local stability results by exploiting the special structure of powers of the Jacobian matrix. Secondly, we show that local stability implies global stability using two different techniques. For one set of parameter values, a contraction mapping result is applied, while for the remaining set of parameter values, we show that the result follows by first proving that the recurrence structure is eventually monotonic in each of its arguments.

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arXiv2020

Chaotic dynamics in a simple predator-prey model with discrete delay

A discrete delay is included to model the time between the capture of the prey and its conversion to viable biomass in the simplest classical Gause type predator-prey model that has equilibrium dynamics without delay. As the delay increases from zero, the coexistence equilibrium undergoes a supercritical Hopf bifurcation, two saddle-node bifurcations of limit cycles, and a cascade of period doublings, eventu1ally leading to chaos. The resulting periodic orbits and the strange attractor resemble their counterparts for the Mackey-Glass equation. Due to the global stability of the system without delay, these complicated dynamics can be solely attributed to the introduction of the delay. Since many models include predator-prey like interactions as submodels, this study emphasizes the importance of understanding the implications of overlooking delay in such models on the reliability of the model-based predictions, especially since the temperature is known to have an effect on the length of certain delays.

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arXiv2020

Growth on multiple interactive-essential resources in a self-cycling fermentor: An impulsive differential equations approach

We introduce a model of the growth of a single microorganism in a self-cycling fermentor in which an arbitrary number of resources are limiting, and impulses are triggered when the concentration of one specific substrate reaches a predetermined level. The model is in the form of a system of impulsive differential equations. We consider the operation of the reactor to be successful if it cycles indefinitely without human intervention and derive conditions for this to occur. In this case, the system of impulsive differential equations has a periodic solution. We show that success is equivalent to the convergence of solutions to this periodic solution. We provide conditions that ensure that a periodic solution exists. When it exists, it is unique and attracting. However, we also show that whether a solution converges to this periodic solution, and hence whether the model predicts that the reactor operates successfully, is initial condition dependent. The analysis is illustrated with numerical examples.

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arXiv2020

A qualitative study of an anaerobic food-web reveals the importance of hydrogen for microbial stability

The mathematical analysis of a three-tiered food-web describing anaerobic chlorophenol mineralisation has suggested the emergence of interesting dynamical behaviour through its specific ecological interactions, which include competition, syntrophy and product inhibition. Previous numerical analyses have revealed the possibility for a Hopf bifurcation occurring through the interior equilibrium and the role of extraneous substrate inputs in both mitigating the emergence of periodic solutions and expanding the desired stable positive steady-state, where full mineralisation occurs. Here we show that, for a generalised model, the inflow of multiple substrates results in greater dynamical complexity and prove the occurrence of a supercritical Hopf bifurcation resulting from variations in these operating parameters. Further, using numerical estimation, we also show that variations in the dilution rate can lead to Bogdanov-Takens and Bautin bifurcations. Finally, we are able to show apply persistence theory for a range of parameter sets to demonstrate unique persistence in the cases where chlorophenol and hydrogen are extraneously added to the system, mirroring recent applied studies highlighting the role of hydrogen in maintaining stable anaerobic microbial communities.

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arXiv2019

A Criterion for the Existence of Relaxation Oscillations with Applications to Predator-Prey Systems and an Epidemic Model

We derive characteristic functions to determine the number and stability of relaxation oscillations for a class of planar systems. Applying our criterion, we give conditions under which the chemostat predator-prey system has a globally orbitally asymptotically stable limit cycle. Also we demonstrate that a prescribed number of relaxation oscillations can be constructed by varying the perturbation for an epidemic model studied by Li et al. [SIAM J. Appl. Math, 2016].

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arXiv2019

Global analysis of a simplified model of anaerobic digestion and a new result for the chemostat

A. Bornhöft, R. Hanke-Rauschenbach, and K. Sundmacher, [Nonlinear Dyn., 73 (2013), pp. 535-549] introduced a qualitative simplification to the ADM1 model for anaerobic digestion. We obtain global results for this model by first analyzing the limiting system, a model of single species growth in the chemostat in which the response function is non-monotone and the species decay rate is included. Using a Lyapunov function argument and the theory of asymptotically autonomous systems, we prove that even in the parameter regime where there is bistability, no periodic orbits exist and every solution converges to one of the equilibrium points. We then describe two algorithms for stochastically perturbing the parameters of the model. Simulations done with these two algorithms are compared with simulations done using the Gillespie and tau-leaping algorithms. They illustrate the severe impact environmental factors may have on anaerobic digestion in the transient phase.

q-bio.QM↗

arXiv2018

Growth on Two Limiting Essential Resources in a Self-Cycling Fermentor

A system of impulsive differential equations with state-dependent impulses is used to model the growth of a single population on two limiting essential resources in a self-cycling fermentor. Potential applications include water purification and biological waste remediation. The self-cycling fermentation process is a semi-batch process and the model is an example of a hybrid system. In this case, a well-stirred tank is partially drained, and subsequently refilled using fresh medium when the concentration of both resources (assumed to be pollutants) falls below some acceptable threshold. We consider the process successful if the threshold for emptying/refilling the reactor can be reached indefinitely without the time between successive emptying/refillings becoming unbounded and without interference by the operator. We prove that whenever the process is successful, the model predicts that the concentrations of the population and the resources converge to a positive periodic solution. We derive conditions for the successful operation of the process that are shown to be initial condition dependent and prove that if these conditions are not satisfied, then the reactor fails. We show numerically that there is an optimal fraction of the medium drained from the tank at each impulse that maximizes the output of the process.

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