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

Infection-induced increases to population size during cycles in a discrete-time epidemic model

One-dimensional discrete-time population models, such as those that involve Logistic or Ricker growth, can exhibit periodic and chaotic dynamics. Expanding the system by one dimension to incorporate epidemiological interactions causes an interesting complexity of new behaviors. Here, we examine a discrete-time two-dimensional susceptible-infectious (SI) model with Ricker growth and show that the introduction of infection can not only produce a distinctly different bifurcation structure than that of the underlying disease-free system but also lead to counter-intuitive increases in population size. We use numerical bifurcation analysis to determine the influence of infection on the location and types of bifurcations. In addition, we examine the appearance and extent of a phenomenon known as the 'hydra effect,' i.e., increases in total population size when factors, such as mortality, that act negatively on a population, are increased. Previous work, primarily focused on dynamics at fixed points, showed that the introduction of infection that reduces fecundity to the SI model can lead to a so-called 'infection-induced hydra effect.' Our work shows that even in such a simple two-dimensional SI model, the introduction of infection that alters fecundity or mortality can produce dynamics can lead to the appearance of a hydra effect, particularly when the disease-free population is at a cycle.

Bubble Identification in the Emerging Economy Fuel Price Series: Evidence from Generalized Sup Augmented Dickey–Fuller Test

In the recent past, the world in general and Pakistan in particular faced a drastic fuel price change, affecting the economic productivity of the country. This has drawn the attention of empirical researchers to analyze the abrupt change in fuel prices. This study takes a lead and investigates for the first time, in the literature related to Pakistan, the presence of multiple fuel price bubbles, with the purpose of knowing if the price driver is due to demand or it is exuberant consumer behavior that prevails and contributes to a sudden boom in fuel price series. The empirical analysis is performed through a recently proposed state-of-the-art generalized sup ADF (GSADF) approach on six commonly used fuel price series, namely, LDO (light diesel oil), HSD (high-speed diesel), petrol, natural gas, kerosene, and MS (motor spirit). The bubble analysis for each of the six fuel price series is based on monthly data from July 2005 to August 2020. The findings provide evidence of the existence of multiple bubbles in all series considered. Specifically, four bubbles are detected in each of the kerosene and natural gas price series, whereas three bubbles are noted in each of the HSD, LDO, petrol and MS price series. The maximum duration of occurrence of bubbles is of 12 months for kerosene. The date-stamping of the bubbles shows that the financial crisis of 2008 contributed to the emergence of bubbles that pushed oil prices upward and caused a depreciation in the national currency.

Disease-Induced Hydra Effect with Overcompensatory Recruitment

In ecological systems, the hydra effect is an increase in population size caused by an increase in mortality. This seemingly counterintuitive effect has been observed in several populations, including fish, blowflies, snails and plants, and has been modeled in both continuous and discrete time. A similar effect induced by disease has recently been observed empirically. Here we present theoretical and simulation results for an infectious disease-induced hydra effect, namely conditions under which the total population size, composed of those that are infectious as well as those that are susceptible, at an endemic equilibrium is greater than the population size at the disease-free equilibrium. (For an endemic k-cycle, this can be similarly defined using the average population.) We find this disease-induced hydra effect occurs when the intra-specific competition is strong and disease infection sufficiently inhibits the reproductive output of infected individuals. For our continuous time model, we give a necessary and sufficient condition for a disease-induced hydra effect. This condition requires overcompensatory recruitment. With a discrete time model, we show there is no disease-induced hydra effect without overcompensatory recruitment. We illustrate by simulations that a disease-induced hydra effect may occur with Ricker recruitment when the endemic system converges to either a fixed equilibrium or a 2-cycle.

Dynamics of a stage-structured predator-prey model: cost and benefit of fear-induced group defense

In the predator-prey system, predators can affect the prey population (1) by direct killing and (2) by inducing predation fear, which ultimately force preys to adopt some anti-predator strategies. However, the anti-predator strategy is not the same for all individual preys of different life stages. Also, anti-predator behavior has both cost and benefit, but most of the mathematical models observed the dynamics by incorporating its cost only. In the present study, we formulate a predator-prey model dividing the prey population into two stages: juvenile and adult. We assume that adult preys are only adapting group defense as an anti-predator strategy when they are sensitive to predation. Group defense plays a positive role for adult prey by reducing their predation, but, on the negative side, it simultaneously decreases their reproductive potential. A parameter, anti-predator sensitivity is introduced to interlink both the benefit and cost of group defense. Our result shows that when adult preys are not showing anti-predator behavior, with an increase of maturation rate, the system exhibits a population cycle of abruptly increasing amplitude, which may drive all species of the system to extinction. Anti-predator sensitivity may exclude oscillation through homoclinic bifurcation and avert the prey population for any possible random extinction. Anti-predator sensitivity also decreases the predator population density and produces bistable dynamics. Higher values of anti-predator sensitivity may lead to the extinction of the predator population and benefit adult preys to persist with large population density. Below a threshold value of anti-predator sensitivity, it may possible to retain the predator population in the system by increasing the fear level of the predator. We also observe our fear-induced stage-structured model exhibits interesting and rich dynamical behaviors, various types of bistabilities in different bi-parameter planes. Finally, we discuss the potential impact of our findings.

Bifurcations and hydra effects in Bazykin's predator-prey model

We consider Bazykin's model to address harvesting induced stability exchanges through bifurcation analysis. We examine the existence of hydra effects and analyze the stock pattern under predator harvesting. Prey harvesting cannot produce hydra effects in our model, whereas predator harvesting may cause multiple hydra effects. Our study reveals that type II response function and mutual interference among predators jointly induce multiple hydra effects and bistability. Bifurcations such as single Hopf-bifurcation, multiple Hopf-bifurcations and multiple saddle-node bifurcations appear for increasing harvesting rate on the predators. However, over-exploitation of the predators cannot generate any such bifurcation in our study. In simulations, the maximum sustainable yield (MSY) exists at a globally stable state. When predator is culled under increasing effort, basin of attraction of the equilibrium corresponding to the higher predator stock gets expanded, which alternatively is in favor of stock benefit for predators. The ecological theory developed in this study might be useful to understand conservation policy and fishery management.

Destabilization and chaos induced by harvesting: insights from one-dimensional discrete-time models

One-dimensional discrete-time population models are often used to investigate the potential effects of increasing harvesting on population dynamics, and it is well known that suitable harvesting rates can stabilize fluctuations of population abundance. However, destabilization is also a possible outcome of increasing harvesting even in simple models. We provide a rigorous approach to study when harvesting is stabilizing or destabilizing, considering proportional harvesting and constant quota harvesting, that are usual strategies for the management of exploited populations. We apply our results to some of the most popular discrete-time population models (quadratic, Ricker and Bellows maps). While the usual case is that increasing harvesting is stabilizing, we prove, somehow surprisingly, that increasing values of constant harvesting can destabilize a globally stable positive equilibrium in some cases; moreover, we give a general result which ensures that global stability can be shifted to observable chaotic dynamics by increasing one model parameter, and apply this result to some of the considered harvesting models.

Threshold harvesting as a conservation or exploitation strategy in population management

Threshold harvesting removes the surplus of a population above a set threshold and takes no harvest below the threshold. This harvesting strategy is known to prevent overexploitation while obtaining higher yields than other harvesting strategies. However, the harvest taken can vary over time, including seasons of no harvest at all. While this is undesirable in fisheries or other exploitation activities, it can be an attractive feature of management strategies where removal interventions are costly and desirable only occasionally. In the presence of population fluctuations, the issue of variable harvests and population sizes becomes even more notorious. Here, we investigate the impact of threshold harvesting on the dynamics of both population size and harvests, especially in the presence of population cycles. We take into account semelparous and iteroparous life cycles, Allee effects, observation uncertainty, and demographic as well as environmental stochasticity, using generic mathematical models in discrete time. Our results show that threshold harvesting enhances multiple forms of population stability, namely persistence, constancy, resilience, and dynamic stability. We discuss plausible choices of threshold values, depending on whether the aim is resource exploitation, pest control, or the stabilization of fluctuations.

The q-deformed Tinkerbell map

q-deformations of functions and distributions have been used in the literature to explain several experimental observations. In this work, we study the dynamics of the Tinkerbell map under q -deformations. The system exhibits a rich variety of dynamical behavior as q varies, including occurrences of interior crises, paired cascades, simultaneous occurrence of Neimark-Sacker and reverse Neimark-Sacker bifurcations, and co-existence of attractors and multistability. Numerical analysis reveals the existence of 3 limit cycles occurring simultaneously in a certain parameter regime. An appropriate choice of initial conditions enables one to choose a desired attractor for the system among other co-existing ones, thus switching the system between different dynamical states. We demonstrate the possibility of secure encryption and decryption of messages with the q -deformed Tinkerbell map. The system's sensitivity to the initial conditions and to the deformation parameter makes the cryptic message secure, and decrypting the original message difficult. We propose the use of the q -deformed map as a novel method for transmission of messages securely.

Hydra effects in discrete-time models of stable communities

A species exhibits a hydra effect when, counter-intuitively, increased mortality of the species causes an increase in its abundance. Hydra effects have been studied in many continuous time (differential equation) multispecies models, but only rarely have hydra effects been observed in or studied with discrete time (difference equation) multispecies models. In addition most discrete time theory focuses on single-species models. Thus, it is unclear what unifying characteristics determine when hydra effects arise in discrete time models. Here, using discrete time multispecies models (where total abundance is the single variable describing each population), I show that a species exhibits a hydra effect in a stable system only when fixing that species' density at its equilibrium density destabilizes the system. This general characteristic is referred to as subsystem instability. I apply this result to two-species models and identify specific mechanisms that cause hydra effects in stable communities, e.g., in host--parasitoid models, host Allee effects and saturating parasitoid functional responses can cause parasitoid hydra effects. I discuss how the general characteristic can be used to identify mechanisms causing hydra effects in communities with three or more species. I also show that the condition for hydra effects at stable equilibria implies the system is reactive (i.e., density perturbations can grow before ultimately declining). This study extends previous work on conditions for hydra effects in single-species models by identifying necessary conditions for stable systems and sufficient conditions for cyclic systems. In total, these results show that hydra effects can arise in many more communities than previously appreciated and that hydra effects were present, but unrecognized, in previously studied discrete time models.

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.

Stabilization of prescribed values and periodic orbits with regular and pulse target oriented control

Investigating a method of chaos control for one-dimensional maps, where the intervention is proportional to the difference between a fixed value and a current state, we demonstrate that stabilization is possible in one of the two following cases: (1) for small values, the map is increasing and the slope of the line connecting the points on the line with the origin is decreasing; (2) the chaotic map is locally Lipschitz. Moreover, in the latter case we prove that any point of the map can be stabilized. In addition, we study pulse stabilization when the intervention occurs each m-th step and illustrate that stabilization is possible for the first type of maps. In the context of population dynamics, we notice that control with a positive target, even if stabilization is not achieved, leads to persistent solutions and prevents extinction in models which experience the Allee effect.

Harvest timing and its population dynamic consequences in a discrete single-species model

The timing of harvesting is a key instrument in managing and exploiting biological populations and renewable resources. Yet, there is little theory on harvest timing, and even less is known about the impact of different harvest times on the stability of population dynamics, even though this may drive population variability and risk of extinction. Here, we employ the framework proposed by Seno to study how harvesting at specific moments in the reproductive season affects not only population size but also stability. For populations with overcompensation, intermediate harvest times tend to be stabilizing (by simplifying dynamics in the case of unimodal maps and by preventing bubbling in the case of bimodal maps). For populations with a strong Allee effect, however, intermediate harvest times can have a twofold effect. On the one hand, they facilitate population persistence (if harvesting effort is low). On the other hand, they provoke population extinction (if harvesting effort is high). Early harvesting, currently considered common sense to take advantage of compensatory effects, may cut into the breeding stock when the population has not yet surpassed the critical Allee threshold. The results in this paper highlight, for the first time, the crucial interplay between harvest timing and Allee effects. Moreover, they demonstrate that harvesting with the same effort but at different moments in time can dramatically alter the impact on the population.