Non-overlapping Generations Research Articles

A discrete-time model of phenotypic evolution

A model is proposed for the phenotypic evolution of a very large population under sustained environmental change and non-overlapping generations, with a single trait considered. Due to an extension of the standard law of quantitative inheritance, each evolutionary mechanism corresponds to a function between random variables associated with distinct stages of the life cycle. Such an approach leads to a two-dimensional map where the dynamics of the phenotypic mean and variance are directly connected. Then, the declining population paradigm is explored in terms of the critical rate of environmental change and using the techniques of the dynamical systems theory. Our results first reveal the opposing pressures on the phenotypic variance due to the conflict between phenotypic load and the ability to pursue the optimum, translated into an optimal value for maximizing the critical rate. Secondly, the introduction of development, through the particular case of linear plasticity, leads to a decreasing degree of stability with the magnitude of plasticity, which means that the recovery time from disturbances is harmed as the plastic effect intensifies, even though no constitutive costs have been assumed, a feature almost as important as the mean fitness to the viability of populations subject to persistent changes. Notwithstanding, the system is stable, and the growth rate benefits from increased plasticity, as expected.

Predator–Prey Interaction with Fear Effects: Stability, Bifurcation and Two-Parameter Analysis Incorporating Complex and Fractal Behavior

This study investigates the dynamics of predator–prey interactions with non-overlapping generations under the influence of fear effects, a crucial factor in ecological research. We propose a novel discrete-time model that addresses limitations of previous models by explicitly incorporating fear. Our primary question is: How does fear influence the stability of predator–prey populations and the potential for chaotic dynamics? We analyze the model to identify biologically relevant equilibria (fixed points) and determine the conditions for their stability. Bifurcation analysis reveals how changes in fear levels and predation rates can lead to population crashes (transcritical bifurcation) and complex population fluctuations (period-doubling and Neimark–Sacker bifurcations). Furthermore, we explore the potential for controlling chaotic behavior using established methods. Finally, two-parameter analysis employing Lyapunov exponents, spectrum, and Kaplan–Yorke dimension quantifies the chaotic dynamics of the proposed system across a range of fear and predation levels. Numerical simulations support the theoretical findings. This study offers valuable insights into the impact of fear on predator–prey dynamics and paves the way for further exploration of chaos control in ecological models.

Population Consequences of Parent-Offspring Competition: An Individual-Based Model

The article compares different versions of individual-based models of single population dynamics with overlapping generations with results of the model of population with non-overlapping generations. In all models, various versions of global competition for common resources and their unequal partitioning between competing individuals are analysed, i.e., when juveniles and adult individuals compete for the same resources, when juveniles and adult individuals use different resources, and a case called mother’s care. The article analyses the relationship between individual variability and population persistence measured by the time of population extinction. Persistence of a population is increased by all forms of competition between individuals not weakened by any environmental factors (e.g. diversification of resources of competing individuals) or factors arising from evolutionary history of the species (e.g. mother’s care) and related in an appropriate way to the mechanism shaping individual variability in the population. Any form of weakening competition leading to decreasing individual variability will adversely affect the persistence of the population. However, differences between versions of the model are not very big.

Turing-like patterns induced by the competition between two stable states in a discrete-time predator–prey model

Patterns, typical spatiotemporal ordered structures, are widely present in a variety of systems, which are the results of the emergence of complex systems. The theory of Turing, occupying a long-range inhibitor and a short-range activator, provides an explanation for the diversity of Turing patterns. Turing-like patterns, caused by various non-Turing mechanisms, have also been investigated in various models. Patterns resulting from Turing or non-Turing mechanisms have predominantly focused on continuous-time models, however, for populations with non-overlapping generations, discrete-time models are more realistic than continuous-time ones. Here, we investigated a type of Turing-like patterns in a discrete-time model. The results show that the competition between two stable states can lead to the formation of Turing-like patterns. Specifically, we obtained a type of Turing-like patterns in a discrete-time model, which exist outside the parameter regions of Turing instability. Only by applying strong impulse noise to the homogeneous stable state can the patterns be excited. And the excitation threshold is related to the control parameter. Analysis shows that the Turing-like patterns are the results of competition between two stable states corresponding to adjacent orbits, and the excitation threshold is determined by the relative position between orbits. This findings exemplify the multitude of mechanisms observed in nature that give rise to the emergence of Turing or Turing-like patterns, and shed light on the pattern formation for Turing/Turing-like patterns.

ANALYSIS OF SKELLAM MODELS WITH A RIGID HARVESTING STRATEGY

Difference equations are used in order to model the dynamics of populations with non-overlapping generations, since the growth of such populations occurs only at discrete points in time. In the simplest case such equations have form $N_{t+1}= F(N_t)$, where $N_t >0$ is the population size at a moment of time $t$, and $F$ is a smooth function. Among such equations the discrete Skellam model are most often used in practice. In the given paper the Skellam model of the form $N_{t+1}=a N_t/ (1+b N_t)$, $N_{t+1}=a N_t^2/ (b^2+ N_t^2)$, $N_{t+1}=a N_t/ (1+b N_t^2)$ is considered, where the parameters $a,b>0$ with taking an effect of harvesting. Positive equilibrium points and conditions for their stability for these equations were found. It is shown in analytical form that these equations do not have periodic solutions with period $T=2$, which means, according by the Sharkovskii theorem, periodic solutions of any periods. In the model with harvesting, only regimes with monotonic stabilization of the population size are observed. Therefore, in all models of Skellam with harvesting, the existence of a critical conception is show, beyond which the population will be completely destroyed. For practice it is important to know the permissible limits of harvesting intensity, which are found in this paper.

The Evolutionary Dynamics of a Sex-Structured Population with Non-Overlapping Generations

This paper proposes and studies a discrete-time model for a sex-structured population with non-overlapping generations under density-dependent regulation of survival. The population is assumed to have genetic variety among individuals in terms of reproductive potential, controlled by a single autosomal diallelic locus. We consider a panmictic population with Mendelian inheritance rules. We examine the stability model and show that increasing the average value of reproductive potential destabilizes the population dynamics. The scenario of stability loss in fixed points via period doubling or Neimark–Sacker bifurcations depends on the intensity of the self-regulation. The growth rate at which the population survives and develops is shown to depend on the fitness of the genotypes and the secondary sex ratio. As a result, the asymptotic genetic composition of the population is determined by the values of the reproductive potentials of the heterozygote and homozygotes, the initial conditions, and the parameter describing the ratio of newborn females to males. With disruptive selection, the influence of external factors changing the current genetic composition of a population can alter the direction of evolution and lead to the extinction of a successful developing population or a gradual population recovery due to evolutionary rescue after a noticeable decline in its abundance.

Ergodicity of the Fisher infinitesimal model with quadratic selection

We study the convergence towards a unique equilibrium distribution of the solutions to a time-discrete model with non-overlapping generations arising in quantitative genetics. The model describes the dynamics of a phenotypic distribution with respect to a multi-dimensional trait, which is shaped by selection and Fisher’s infinitesimal model of sexual reproduction. We extend some previous works devoted to the time-continuous analogs, that followed a perturbative approach in the regime of weak selection, by exploiting the contractivity of the infinitesimal model operator in the Wasserstein metric. Here, we tackle the case of quadratic selection by a global approach. We establish uniqueness of the equilibrium distribution and exponential convergence of the renormalized profile. Our technique relies on an accurate control of the propagation of information across the large binary trees of ancestors (the pedigree chart), and reveals an ergodicity property, meaning that the shape of the initial datum is quickly forgotten across generations. We combine this information with appropriate estimates for the emergence of Gaussian tails and propagation of quadratic and exponential moments to derive quantitative convergence rates. Our result can be interpreted as a generalization of the Krein–Rutman theorem in a genuinely non-linear, and non-monotone setting.

Move it or lose it: Predicted effects of culverts and population density on Mojave desert tortoise (Gopherus agassizii) connectivity.

Roadways and railways can reduce wildlife movements across landscapes, negatively impacting population connectivity. Connectivity may be improved by structures that allow safe passage across linear barriers, but connectivity could be adversely influenced by low population densities. The Mojave desert tortoise is threatened by habitat loss, fragmentation, and population declines. The tortoise continues to decline as disturbance increases across the Mojave Desert in the southwestern United States. While underground crossing structures, like hydrological culverts, have begun receiving attention, population density has not been considered in tortoise connectivity. Our work asks a novel question: How do culverts and population density affect connectivity and potentially drive genetic and demographic patterns? To explore the role of culverts and population density, we used agent-based spatially explicit forward-in-time simulations of gene flow. We constructed resistance surfaces with a range of barriers to movement and representative of tortoise habitat with anthropogenic disturbance. We predicted connectivity under variable population densities. Simulations were run for 200 non-overlapping generations (3400 years) with 30 replicates using 20 microsatellite loci. We evaluated population genetic structure and diversity and found that culverts would not entirely negate the effects of linear barriers, but gene flow improved. Our results also indicated that density is important for connectivity. Low densities resulted in declines regardless of the landscape barrier scenario (> 75% population census size, > 97% effective population size). Results from our simulation using current anthropogenic disturbance predicted decreased population connectivity over time. Genetic and demographic effects were detectable within five generations (85 years) following disturbance with estimated losses in effective population size of 69%. The pronounced declines in effective population size indicate this could be a useful monitoring metric. We suggest management strategies that improve connectivity, such as roadside fencing tied to culverts, conservation areas in a connected network, and development restricted to disturbed areas.

Determining the most recent common ancestor in a finite linear habitat with asymmetric dispersal

Many species that are birthed in one location and become reproductive in another location can be treated as if in a one-dimensional habitat where dispersal is biased downstream. One example of such is planktonic larvae that disperse in coastal oceans, rivers, and streams. In these habitats, the dynamics of the dispersal are dominated by the movement of offspring in one direction and the distance between parents and offspring in the other direction does not matter. We study an idealized species with non-overlapping generations in a finite linear habitat that has no larval input from outside of the habitat and is therefore isolated from other populations. The most non-realistic assumption that we make is that there are non-overlapping generations, and this is an assumption to be considered in future work. We find that a biased dispersal in the habitat reduces the average time to the most recent common ancestor and causes the average location of the most recent common ancestor to move from the center of the habitat to the upstream edge of the habitat. Due to the decrease in the time to the most recent common ancestor and the shift of the average location to the upstream edge, the effective population size (Ne) no longer depends on the census size and is dependent on the dispersal statistics. We determine the average time and location of the most recent common ancestor as a function of the larval dispersal statistics. The location of the most recent common ancestor becomes independent of the length of the habitat and is only dependent on the location of the upstream edge and the larval dispersal statistics.

Branching with Selection and Mutation I: Mutant Fitness of Fréchet Type

We investigate two stochastic models of a growing population with discrete and non-overlapping generations, subject to selection and mutation. In our models each individual carries a fitness which determines its mean offspring number. Many of these offspring inherit their parent’s fitness, but some are mutants and obtain a fitness randomly sampled, as in Kingman’s house-of-cards model, from a distribution in the domain of attraction of the Fréchet distribution. We give a rigorous proof for the precise rate of superexponential growth of these stochastic processes and support the argument by a heuristic and numerical study of the mechanism underlying this growth. This study yields in particular that the empirical fitness distribution of one model in the long time limit displays periodic behaviour.

Numerical dynamics of integrodifference equations

Integrodifference equations are versatile models in theoretical ecology for the spatial dispersal of species evolving in non-overlapping generations. The dynamics of these infinite-dimensional discrete dynamical systems is often illustrated using computational simulations. This paper studies the effect of Nyström discretization to the local dynamics of periodic integrodifference equations having Hölder continuous functions over a compact domain as state space. We prove persistence and convergence for hyperbolic periodic solutions and their associated stable and unstable manifolds respecting the convergence order of the quadrature/cubature method.

Bifurcation analysis of a discrete type prey-predator model with Michaelis–Menten harvesting in predator

Abstract A discrete predator–prey model with square root functional response describing prey herd behavior and nonlinear predator harvesting has been considered in the present work. Three equilibria of the system have been found and observed that two equilibrium points always exist and are feasible, but the interior equilibrium point is feasible under a parametric condition. The local stability of the three equilibria has been analyzed. The interior equilibrium point is locally asymptotically stable under a parametric condition. It is examined that a flip and Neimark–Sacker bifurcations have occurred in the system at the axial equilibrium point. The flip and Neimark–Sacker bifurcations have been analyzed by the center manifold theorem and bifurcation theory, considering the harvesting coefficient as the bifurcation parameter. The proposed discrete model with a nonlinear Michaelis–Menten type harvesting effect on the predator population exhibits rich dynamics; for instance, bifurcations, chaos, and more complex dynamical behaviors. The discrete-time model also produced few numerical simulation results that are more accurate than the continuous model. The proposed discrete model will be performed better than the continuous model in populations with non-overlapping generations or smaller densities. The harvesting coefficient’s optimal value has finally been identified, and an optimal harvesting policy has been introduced. To verify the results, further numerical simulations have been performed.

Complex dynamics and chaos control in a nonlinear discrete prey–predator model

The dynamics of prey–predator interactions are often modeled using differential or difference equations. In this paper, we investigate the dynamical behavior of a two-dimensional discrete prey–predator system. The model is formulated in terms of difference equations and derived by using a nonstandard finite difference scheme (NSFD), which takes into consideration the non-overlapping generations. The existence of fixed points as well as their local asymptotic stability are proved. Further, it is shown that the model experiences Neimark–Sacker bifurcation (NSB for short) and period-doubling bifurcation (PDB) in a small neighborhood of the unique positive fixed point under certain parametric conditions. This analysis utilizes bifurcation theory and the center manifold theorem. The chaos produced by NSB and PDB is stabilized. Finally, we use numerical simulations and computer analysis to check our theories and show more complex behaviors.

Divergence of a speciation trait through artificial selection: Insights into constraints, by-product effects and sexual isolation.

Speciation and sexual isolation often occur when divergent female mating preferences target male secondary sexual traits. Despite the importance of such male signals, little is known about their evolvability and genetic linkage to other traits during speciation. To answer these questions, we imposed divergent artificial selection for 10 non-overlapping generations on the Inter-Pulse-Interval (IPI) of male courtship songs; which has been previously shown to be a major species recognition trait for females in the Drosophila athabasca species complex. Focusing on one of the species, Drosophila mahican (previously known as EA race), we examined IPI's: (1) rate of divergence, (2) response to selection in different directions, (3) genetic architecture of divergence and (4) by-product effects on other traits that have diverged in the species complex. We found rapid and consistent response for higher IPI but less response to lower IPI; implying asymmetrical constraints. Genetic divergence in IPI differed from natural species in X versus autosome contribution and in dominance, suggesting that evolution may take different paths. Finally, selection on IPI did not alter other components of male songs, or other ecological traits, and did not cause divergence in female preferences, as evidenced by lack of sexual isolation. This suggests that divergence of male courtship song IPI is unconstrained by genetic linkage with other traits in this system. This lack of linkage between male signals and other traits implies that female preferences or ecological selection can co-opt and mould specific male signals for species recognition free of genetic constraints from other traits.

36 Exploring the Statistical Nature of Independent Chromosome Segments

Abstract Independent chromosome segments for a population with Ne effective population size and L genome length can be approximately defined as 4NeL non-overlapping haplotypes of L/2 length derived from any Ne animals. The number of independent chromosome segments (Me) can be approximated as 4NeL. The genetic selection with a genomic relationship matrix (GRM) using such haplotypes approaches that with a GRM using the SNP markers. The objective of this study was to investigate the statistical nature of independent chromosome segments. Data were simulated using QMSim and contained a population of ten non-overlapping generations, each including 2,000 animals with Ne equal to 20, and a polygenic trait with a heritability of 0.6. The last three generations were genotyped, and each genome contained ten 1 M long chromosomes, for a total genome length of 10 M and 50,000 SNP. Chromosome segments of each animal were organized in non-overlapping haplotypes by the SNP code using in-house software written in Fortran. The effects of the hypothetical independent chromosome segments were estimated using a model that included an overall mean plus the segment effects. To analyze the behavior around Me, the number of segments chosen to estimate segment effects varied around 4NeL. Accuracies were calculated for animals in the last generation by cor (TBV, Zsŝ), where Zs ŝ is a vector of breeding values based on segments and TBV is a vector of the true breeding values outputted by QMSim. Accuracies of segment effects were compared with the true accuracy, i.e., cor (TBV,GEBV). The maximum accuracy of segment effects was 0.84, and the true accuracy was 0.96. The results suggest that 4NeL segments contain most of the additive information in the population. However, the accuracy of GBLUP is greater than that of the chromosome segment effects, suggesting that arranging chromosome segments based solely on the statistical nature is not enough to account for all the genetic variation.

Carry-over effects and fitness trade-offs in marine life histories: The costs of complexity for adaptation.

Most marine organisms have complex life histories, where the individual stages of a life cycle are often morphologically and ecologically distinct. Nevertheless, life-history stages share a single genome and are linked phenotypically (by "carry-over effects"). These commonalities across the life history couple the evolutionary dynamics of different stages and provide an arena for evolutionary constraints. The degree to which genetic and phenotypic links among stages hamper adaptation in any one stage remains unclear and yet adaptation is essential if marine organisms will adapt to future climates. Here, we use an extension of Fisher's geometric model to explore how both carry-over effects and genetic links among life-history stages affect the emergence of pleiotropic trade-offs between fitness components of different stages. We subsequently explore the evolutionary trajectories of adaptation of each stage to its optimum using a simple model of stage-specific viability selection with nonoverlapping generations. We show that fitness trade-offs between stages are likely to be common and that such trade-offs naturally emerge through either divergent selection or mutation. We also find that evolutionary conflicts among stages should escalate during adaptation, but carry-over effects can ameliorate this conflict. Carry-over effects also tip the evolutionary balance in favor of better survival in earlier life-history stages at the expense of poorer survival in later stages. This effect arises in our discrete-generation framework and is, therefore, unrelated to age-related declines in the efficacy of selection that arise in models with overlapping generations. Our results imply a vast scope for conflicting selection between life-history stages, with pervasive evolutionary constraints emerging from initially modest selection differences between stages. Organisms with complex life histories should also be more constrained in their capacity to adapt to global change than those with simple life histories.

On the qualitative study of a two-trophic plant-herbivore model.

The coexistence of plant-herbivore populations in an ecological system is a fundamental topic of research in mathematical ecology. Plant-herbivore interactions are often described by using discrete-time models in the case of non-overlapping generations: such generations have some specific time interval of life and their old generations are replaced by new generations after some regular interval of time. Keeping in mind the dynamical reliability of continuous-time models we presented two discrete-time plant-herbivore models. Mainly, by applying Euler's forward method a discrete-time plant-herbivore model is obtained from a continuous-time plant-herbivore model. In addition, a dynamically consistent discrete-time plant-herbivore model is obtained by applying a nonstandard difference scheme. Moreover, local stability is discussed and the existence of bifurcation about positive equilibrium is shown under some mathematical conditions. To control bifurcation and chaos, a modified hybrid technique is developed. Finally, to support our theocratical results and to show the dynamical reliability of the nonstandard difference scheme some numerical examples are provided.

A critical examination of models of annual‐plant population dynamics and density‐dependent fecundity

Abstract Mathematical models serve many purposes in biology. Each and every model is a necessary simplification of reality, and, as simplifications, these models are also wrong by definition. And yet there are many ways to be wrong, and some of these are a much greater concern than others. For example, a paradoxical model‐based prediction may simply be puzzling whereas unphysical variables (e.g. negative amounts of time or temperatures below absolute zero) and nonbiological variables (e.g. negative abundances or negative feeding rates) should be avoided altogether. Here I analyse a discrete‐time model of annual‐plant population dynamics and three phenomenological models for density‐dependent fecundity. These phenomenological models are generally interchanged solely on the basis of their statistical fit to data. On the other hand, I highlight ways in which their phenomenological basis hampers our ability to capture known aspects of annual‐plant biology. I then demonstrate how to specify a more flexible, biologically appropriate model and illustrate this model's behaviour and interpretation in single‐species and multi‐species contexts. By constructing this generative model for annual‐plant population dynamics, I can also demonstrate how and why emergent phenomena, such as negative density dependence, emerge. Although my focus is on a model applied to annual plants, the biological implications extend to modelling any species with a discrete life cycle and nonoverlapping generations. More broadly, my exploration here showcases that there are many more criteria with which we could, and arguably should, ground‐truth mathematical models across biology.

Qualitative behavior of a discrete predator–prey system under fear effects

Abstract Numerous field data and experiments on the perching birds or songbirds show that the fear of predators can cause significant changes in the prey population. Fear of predatory populations increases the chances of survival of the prey population, and this can greatly reduce the reproduction of the prey population. The influence of fear has contributed a leading role in both the environmental biology and theoretical ecology. Taking into account the interaction of predator–prey with non-overlapping generations, a discrete-time model is proposed and studied. Keeping in mind the biological feasibility of species, the existence of fixed points is studied along with the local asymptotic behavior of the proposed model around these fixed points. Furthermore, taking into account the oscillatory behavior of the model, various types of bifurcations are analyzed about biologically feasible fixed points with an application of center manifold theory and bifurcation theory of normal forms. Existence of chaos is discussed, and fluctuating and chaotic behavior of the system is controlled through implementation of different chaos control procedures. The illustration of theoretical discussion is carried out via validation of observed experimental field data and appropriate numerical simulation.

Simple and complex dynamics in the model of evolution of two populations coupled by migration with non-overlapping generations

Purpose is to study the mechanisms leading to genetic divergence (stable genetic differences between two adjacent populations). We considered the following classical model situation. Populations are panmictic with Mendelian rules of inheritance. The action of natural selection (differences in fitness) on each of population is the same and is determined by the genotypes of only one diallel locus. We assume that adjacent generations do not overlap and genetic transformations can be described by a discrete time model. This model describes the change in the concentration of one of the alleles in each population and the ratio (weight) of first population to the total size. Methods. We used the analogue of saddle charts to construct parametric portraits showing the domains of qualitatively different dynamic modes. The study is supplemented with phase portraits, basins of attraction and bifurcation diagrams. Results. We found that the model dynamic regimes qualitatively coincide with the regimes of a similar model with continuous time, but only for a weak migration. With a strong coupling, fluctuations of the phase variables are possible. We showed that the genetic divergence is possible only with reduced fitness of heterozygotes and is the result of a series of bifurcations: pitchfork bifurcation, period doubling, or saddle-node bifurcation. After these qualitative changes, the dynamics become bi- or quadstable. In the first case, the solutions corresponding to the genetic divergence are unstable and are just a part of the transient process to monomorphic state. In the second case, the divergence is stable and appears as 2-cycle for a strong migration coupling. Conclusion. In neighboring populations, movement towards an asymptotic genetic structure (monomorphism, polymorphism or divergence) can be strictly monotonous or in the form of damped unstable or undamped stable fluctuations with a period of 2 for biologically significant parameters. For insignificant parameters, we found a complex dynamics (chaos) that consist of divergent fluctuations around fixed points and quasi-random transitions between them.