Fig 6 - uploaded by James Mallet
Content may be subject to copyright.
Approximately pure frequency-dependent selection (equations 10 and A2.4). (A) No change in equilibrium density among haplotypes, but a temporary rise in population density occurs during replacement: r 1 = 0.15, α 11 = 0.00015, r 2 = 0.02, α 22 = 0.00002, and γ 12 = 1/ γ 21 = 0.6. (B) No change in equilibrium density among haplotypes, but a temporary fall in population density occurs during
Source publication
Question: Population ecology and population genetics are treated separately in most textbooks. However, Darwin's term the 'struggle for existence' included both natural selection and ecological competition. Using the simplest possible mathematical models, this paper searches for historical reasons for the lack of unity in ecological and evolutionar...
Contexts in source publication
Context 1
... density-independent and frequency-independent selection ( ‘ constant selection ’ ; Fig. 4B, D) can result in density-regulated populations if the per capita density-dependent effects of each species on the other are the same as on its own species (i.e. if α 11 = α 21 and α 22 = α 12 ) (Appendix 2, case 2). The gene frequency trajectory is identical to that under exponential growth (cf. Fig. 4C, D), whether the population is near to or far from equilibrium (Fig. 4B). More generally, these conditions will not pertain, and natural selection will be both frequency-dependent and density-dependent (Smouse, 1976) . However, the density-independent effects of r 1 and r 2 are in essence the ‘ main effects ’ , and the effects of the α parameters are second-order, ‘ interaction effects ’ between haplotypes; it can be envisaged that differences in α values will tend to be smaller between haplotypes than the differences due to intrinsic birth and death rates that are components of r -values. If so, most evolution by natural selection within species may indeed be approximately independent of density and frequency. Although equation (10) is relatively simple, a total of six parameters control population regulation and the strength of natural selection for two haplotypes, so the behaviour is correspondingly rich. Even this most basic, haploid model of natural selection in density- regulated populations can produce anything from constant selection (Fig. 4B, D), to frequency- and density-dependent natural selection, as well as ‘ hard ’ (Fig. 5) and, approximately also, ‘ soft ’ selection (Fig. 6) (Table 1, Appendix 2). Selection may increase or decrease equilibrium density (Fig. 5) or leave it relatively unaffected (Fig. 6), depending on parameter values. Non-linear evolution of logit frequency (Figs. 5, 6) provides a useful definition of frequency-dependent selection. Also, I follow Christiansen (1975) in referring to soft selection as selection that does not alter population density; hard selection is selection that alters population density. Internal equilibria may exist, but stable polymorphism in a haploid model requires frequency-dependent selection; neither density-independent selection, nor density-dependent selection alone are sufficient to allow stable polymorphisms (Table 1, Appendix 2). The above results apply only to the simplest clonal organisms. In sexual diploids, there can be genic selection equivalent to haploid selection provided demographic parameters of heterozygotes are intermediate (Kimura, 1978) . More generally, selection involves competition among sexual diploids: at its simplest among three genotypes at a biallelic locus, A 1 A 1 , A 1 A 2 , and A 2 A 2 . In haploids, the only interactions were α parameters among haplotypes or alleles within populations. Diploidy introduces new potential interactions among mating individuals, as well as intra-genotype interactions due to dominance and heterosis, and variation among progenies of each genotype (Smouse, 1976) . Births and deaths occur at different times during the life cycle, and can be affected by density dependence differently. All of these combine in complex ways with the purely competitive interactions discussed above for haploids (Appendix 3). The difficulty of analysing diploid models has been exacerbated still further by opaqueness introduced by the r-K logistic formulation, and in many cases by additional complexity due to discrete generations. To save space, I briefly discuss earlier results for sexual diploids but do not deal with the mathematics. Selection among diploids has been explored both in continuous-time models (Kostitzin, 1939; MacArthur, 1962; Smouse, 1976; Desharnais and Costantino, 1983) , and in discrete-time Wrightian models (Anderson, 1971; Charlesworth, 1971; Roughgarden, 1971; Asmussen and Feldman, 1977; Anderson and Arnold, 1983; Asmussen, 1983b) . Since weak selection in discrete generations can be approximated by a continuous-time equivalent, I here ignore the additional paradoxes and chaotic behaviour introduced by discrete time (Cook, 1965; Charlesworth, 1971; May, 1974; Asmussen and Feldman, 1977; Asmussen, 1983b) . These problems do not occur under weak selection and can be eradicated by using arguably more appropriate discrete time formulations (e.g. equation 20 in Asmussen and Feldman, 1977) . In MacArthur ’ s original analysis (MacArthur, 1962) , genotypes were assumed to differ in values of K , but selection depended only on total density of all three genotypes N = n 11 + n 12 + n 22 equivalent in a Lotka-Volterra formulation to the assumption that γ ij , kl = 1 ∀ ij ≠ kl (where ij and kl each represent one of the diploid genotypes 11, 12, 22 in a diploid version of equation 6; see Appendix 3). This is purely density-dependent selection, equivalent to MacArthur and Wilson ’ s K -selection (MacArthur and Wilson, 1967) . Unlike the equivalent haploid/asexual model (Appendix 2, case 3), polymorphic equilibria are possible with selection on diploids. If heterozygotes are intermediate, K 11 ≥ K 12 > K 22 , then haplotype 1 replaces haplotype 2 (Fig. 7A, C), and vice versa for opposite signs. If there is heterosis (over-dominance, or higher fitness of heterozygotes) for K , i.e. K 11 < K 12 > K 22 , stable polymorphisms result at equilibrium (Fig. 7B, D; the reverse inequality, under- dominance, gives an unstable polymorphic equilibrium) (Kostitzin, 1936, 1939; MacArthur, 1962) . Equivalent heterosis under the r - α model, the diploid equivalent of equation (7), can be obtained by noting that K ij = r ij / α ij , ij (Appendix 3). Purely competitive equilibria or declines in population size are not possible when γ ij , kl = 1 ∀ ij ≠ kl as in MacArthur ’ s analysis (Appendix 2, case 3 for haploids), and in most other treatments of diploid density- dependent dynamics (Anderson, 1971; Roughgarden, 1971; Asmussen, 1983b) . On relaxing the pure density-dependent assumption, much richer behaviour emerges, with combinations such as heterozygous advantage accompanied by equilibrium population decline possible (Fig. 7B, D). In her analysis of a discrete-time analogue, Asmussen (1983b) found up to four possible interior equilibria, of which two could be stable; however, complete analytical results were impossible to obtain. The history of these discoveries is somewhat obscure, and is not found in textbooks. When writing their influential population genetics textbook, Crow and Kimura (1970) were concerned to justify classical population genetical constant-selection models in terms of density-dependent demography (Kimura and Crow, 1969; Crow and Kimura, 1970) . The problem with doing this was that r-K Lotka-Volterra ...
Context 2
... density-independent and frequency-independent selection ( ‘ constant selection ’ ; Fig. 4B, D) can result in density-regulated populations if the per capita density-dependent effects of each species on the other are the same as on its own species (i.e. if α 11 = α 21 and α 22 = α 12 ) (Appendix 2, case 2). The gene frequency trajectory is identical to that under exponential growth (cf. Fig. 4C, D), whether the population is near to or far from equilibrium (Fig. 4B). More generally, these conditions will not pertain, and natural selection will be both frequency-dependent and density-dependent (Smouse, 1976) . However, the density-independent effects of r 1 and r 2 are in essence the ‘ main effects ’ , and the effects of the α parameters are second-order, ‘ interaction effects ’ between haplotypes; it can be envisaged that differences in α values will tend to be smaller between haplotypes than the differences due to intrinsic birth and death rates that are components of r -values. If so, most evolution by natural selection within species may indeed be approximately independent of density and frequency. Although equation (10) is relatively simple, a total of six parameters control population regulation and the strength of natural selection for two haplotypes, so the behaviour is correspondingly rich. Even this most basic, haploid model of natural selection in density- regulated populations can produce anything from constant selection (Fig. 4B, D), to frequency- and density-dependent natural selection, as well as ‘ hard ’ (Fig. 5) and, approximately also, ‘ soft ’ selection (Fig. 6) (Table 1, Appendix 2). Selection may increase or decrease equilibrium density (Fig. 5) or leave it relatively unaffected (Fig. 6), depending on parameter values. Non-linear evolution of logit frequency (Figs. 5, 6) provides a useful definition of frequency-dependent selection. Also, I follow Christiansen (1975) in referring to soft selection as selection that does not alter population density; hard selection is selection that alters population density. Internal equilibria may exist, but stable polymorphism in a haploid model requires frequency-dependent selection; neither density-independent selection, nor density-dependent selection alone are sufficient to allow stable polymorphisms (Table 1, Appendix 2). The above results apply only to the simplest clonal organisms. In sexual diploids, there can be genic selection equivalent to haploid selection provided demographic parameters of heterozygotes are intermediate (Kimura, 1978) . More generally, selection involves competition among sexual diploids: at its simplest among three genotypes at a biallelic locus, A 1 A 1 , A 1 A 2 , and A 2 A 2 . In haploids, the only interactions were α parameters among haplotypes or alleles within populations. Diploidy introduces new potential interactions among mating individuals, as well as intra-genotype interactions due to dominance and heterosis, and variation among progenies of each genotype (Smouse, 1976) . Births and deaths occur at different times during the life cycle, and can be affected by density dependence differently. All of these combine in complex ways with the purely competitive interactions discussed above for haploids (Appendix 3). The difficulty of analysing diploid models has been exacerbated still further by opaqueness introduced by the r-K logistic formulation, and in many cases by additional complexity due to discrete generations. To save space, I briefly discuss earlier results for sexual diploids but do not deal with the mathematics. Selection among diploids has been explored both in continuous-time models (Kostitzin, 1939; MacArthur, 1962; Smouse, 1976; Desharnais and Costantino, 1983) , and in discrete-time Wrightian models (Anderson, 1971; Charlesworth, 1971; Roughgarden, 1971; Asmussen and Feldman, 1977; Anderson and Arnold, 1983; Asmussen, 1983b) . Since weak selection in discrete generations can be approximated by a continuous-time equivalent, I here ignore the additional paradoxes and chaotic behaviour introduced by discrete time (Cook, 1965; Charlesworth, 1971; May, 1974; Asmussen and Feldman, 1977; Asmussen, 1983b) . These problems do not occur under weak selection and can be eradicated by using arguably more appropriate discrete time formulations (e.g. equation 20 in Asmussen and Feldman, 1977) . In MacArthur ’ s original analysis (MacArthur, 1962) , genotypes were assumed to differ in values of K , but selection depended only on total density of all three genotypes N = n 11 + n 12 + n 22 equivalent in a Lotka-Volterra formulation to the assumption that γ ij , kl = 1 ∀ ij ≠ kl (where ij and kl each represent one of the diploid genotypes 11, 12, 22 in a diploid version of equation 6; see Appendix 3). This is purely density-dependent selection, equivalent to MacArthur and Wilson ’ s K -selection (MacArthur and Wilson, 1967) . Unlike the equivalent haploid/asexual model (Appendix 2, case 3), polymorphic equilibria are possible with selection on diploids. If heterozygotes are intermediate, K 11 ≥ K 12 > K 22 , then haplotype 1 replaces haplotype 2 (Fig. 7A, C), and vice versa for opposite signs. If there is heterosis (over-dominance, or higher fitness of heterozygotes) for K , i.e. K 11 < K 12 > K 22 , stable polymorphisms result at equilibrium (Fig. 7B, D; the reverse inequality, under- dominance, gives an unstable polymorphic equilibrium) (Kostitzin, 1936, 1939; MacArthur, 1962) . Equivalent heterosis under the r - α model, the diploid equivalent of equation (7), can be obtained by noting that K ij = r ij / α ij , ij (Appendix 3). Purely competitive equilibria or declines in population size are not possible when γ ij , kl = 1 ∀ ij ≠ kl as in MacArthur ’ s analysis (Appendix 2, case 3 for haploids), and in most other treatments of diploid density- dependent dynamics (Anderson, 1971; Roughgarden, 1971; Asmussen, 1983b) . On relaxing the pure density-dependent assumption, much richer behaviour emerges, with combinations such as heterozygous advantage accompanied by equilibrium population decline possible (Fig. 7B, D). In her analysis of a discrete-time analogue, Asmussen (1983b) found up to four possible interior equilibria, of which two could be stable; however, complete analytical results were impossible to obtain. The history of these discoveries is somewhat obscure, and is not found in textbooks. When writing their influential population genetics textbook, Crow and Kimura (1970) were concerned to justify classical population genetical constant-selection models in terms of ...
Context 3
... when density remains the same during evolution, we see a two-phase evolutionary process on a logit scale (equation A2.2, Fig. 6, discussed below). Thus selection is frequency- dependent. Overall density N remains approximately constant during replacement evolution, although ‘ blips ’ in population density can occur when selection is strong (Fig. 6). Pure frequency-dependent selection is both effectively density-independent and frequency- dependent. Furthermore, because equilibrium density does not change, selection can be approximately ‘ soft ’ . Frequency-dependent selection emerges from these simple demographic models solely via differences in α 12 and α 21 interaction among haplotypes. Interior equilibria are possible, both stable and unstable, and are given by the same conditions as for Lotka-Volterra competition (Fig. 2). A special case of interior equilibrium in equation (A2.4) occurs if ˆ = ( α − α 12 )/[( α − α 21 ) + ( α − α 12 )]. Evolution towards or away from interior equilibria will, however, give hard selection, since overall density at equilibrium will be higher than K for stable equilibria, or lower for unstable equilibria, even when K does not differ among haplotypes. For the most general kinds of selection, evolution under equation (10) will consist of two approximately logit-linear phases (Smouse and Kosuda, 1977) ; when haplotype 1 is rare, n 1 ≈ 0, and if the population is at approximate density equilibrium, n 2 ≈ K 2 (as in Fig. 3), then the rate of evolution will ...
Context 4
... when density remains the same during evolution, we see a two-phase evolutionary process on a logit scale (equation A2.2, Fig. 6, discussed below). Thus selection is frequency- dependent. Overall density N remains approximately constant during replacement evolution, although ‘ blips ’ in population density can occur when selection is strong (Fig. 6). Pure frequency-dependent selection is both effectively density-independent and frequency- dependent. Furthermore, because equilibrium density does not change, selection can be approximately ‘ soft ’ . Frequency-dependent selection emerges from these simple demographic models solely via differences in α 12 and α 21 interaction among haplotypes. Interior equilibria are possible, both stable and unstable, and are given by the same conditions as for Lotka-Volterra competition (Fig. 2). A special case of interior equilibrium in equation (A2.4) occurs if ˆ = ( α − α 12 )/[( α − α 21 ) + ( α − α 12 )]. Evolution towards or away from interior equilibria will, however, give hard selection, since overall density at equilibrium will be higher than K for stable equilibria, or lower for unstable equilibria, even when K does not differ among haplotypes. For the most general kinds of selection, evolution under equation (10) will consist of two approximately logit-linear phases (Smouse and Kosuda, 1977) ; when haplotype 1 is rare, n 1 ≈ 0, and if the population is at approximate density equilibrium, n 2 ≈ K 2 (as in Fig. 3), then the rate of evolution will ...
Citations
... This concept is referred to as the carrying capacity (K) of an environment. Carrying (density) capacity has been defined as a population's equilibrium density, which means that the realized rate of per capita increase at the time is 0. The following logistic equation is the most common formulation for the description of population growth and the relationship between r and K [12]: ...
... where N is the insect density at time t, K is the carrying capacity (the same unit as N), and r is the innate rate of increase [12]. Equation (5) indicates that r = dN Ndt when N is very small (much smaller than K), and high values of N tend to have lower values of r. ...
The determination of innate rate of increase (r) values under different grain storage conditions is critical for insect population predictions. The r values for Cryptolestes ferrugineus (Stephens) and Tribolium castaneum (Herbst) were calculated by using a new suggested method (continuous time analysis) and data from the literature, and these calculated r values were compared to identify the r values and carrying capacities under real grain storage conditions and times. The insects were reared in small glass vials (0.3 kg wheat), small PVC columns (2 kg wheat), large PVC columns (14 kg wheat), and shallow containers (14 kg wheat or wheat + cracked wheat). The wheat or cracked wheat had 13.8 to 14.5% moisture contents at different constant temperatures (17.5 to 42.5 °C) and fluctuating temperatures. The r values at the beginning of the population were the highest. Before r became negative, it gradually decreased with increasing time. After the r value became negative, it sometimes increased to positive; however, the rebounded r was much less than the initial r and gradually tended to stabilize within an up-and-down range. This up-and-down r was related to the carrying capacity. The larger the grain bulk, the higher the innate rate was for both species. The r values associated with 14 kg of wheat could be used to predict the insect population dynamics in stored grain bins.
... trait syndromes) is still often overlooked ( [35], but see [40,41]). Finally, when studying demography, it is important to include density dependence as this is a key aspect of population growth and is known to be affected by environmental change drivers [42][43][44]. ...
... When the demographic response was summarized as carrying capacity, however, we did not find such a link. This could suggest that using carrying capacity as a summarizing (derived) parameter for density dependent demography is too simplistic, emphasizing the importance to include density dependent growth and interactions when studying demography [42,43]. ...
Background
Over the past decade, theory and observations have suggested intraspecific variation, trait-based differences within species, as a buffer against biodiversity loss from multiple environmental changes. This buffering effect can only occur when different populations of the same species respond differently to environmental change. More specifically, variation of demographic responses fosters buffering of demography, while variation of trait responses fosters buffering of functioning. Understanding how both responses are related is important for predicting biodiversity loss and its consequences. In this study, we aimed to empirically assess whether population-level trait responses to multiple environmental change drivers are related to the demographic response to these drivers. To this end, we measured demographic and trait responses in microcosm experiments with two species of ciliated protists. For three clonal strains of each species, we measured responses to two environmental change drivers (climate change and pollution) and their combination. We also examined if relationships between demographic and trait responses existed across treatments and strains.
Results
We found different demographic responses across strains of the same species but hardly any interactive effects between the two environmental change drivers. Also, trait responses (summarized in a survival strategy index) varied among strains within a species, again with no driver interactions. Demographic and trait responses were related across all strains of both species tested in this study: Increasing intrinsic growth and self-limitation were associated with a shift in survival strategy from sit-and-wait towards flee.
Conclusions
Our results support the existence of a link between a population’s demographic and trait responses to environmental change drivers in two species of ciliate. Future work could dive deeper into the specifics of phenotypical trait values, and changes therein, related to specific life strategies in different species of ciliate and other zooplankton grazers.
... So it is hardly reasonable to assume a linear decline of per-capita growth rate (pgr) via an instantaneous response to current density, which is the basic assumption of the logistic equation. The density-pgr relationship is preferable to be skewed over the standard logistic setup (Schoener, 1973;Mallet, 2012;Roughgarden, 1997). Logistic growth can rarely be observed in real-life phenomena, and hence unrealistic. ...
... One of the assumptions in our study is that the SGC costs only affect rates of exponential growth. It is possible, however, that SGC affects other characteristics of bacterial growth, for example, allowing bacteria to achieve different maximal densities, causing a correlation between r and K (25). A correlation pattern between maximal density and the average growth rate of both competitors would indicate competitions with fast-growing strains predicted effect of epistasis on the growth rate of a genotype. ...
In this article, we present a method for designing, executing, and analyzing data from a microbial competition experiment. We use fluorescent reporters to label different competing strains and resolve individual growth curves using a fluorescent spectrophotometer. Our comprehensive data analysis pipeline integrates multiple experiments to simultaneously infer sources of variation, extract selection coefficients, and estimate the genetic contributions to fitness for various synthetic genetic cassettes (SGCs). To demonstrate the method, we employ a synthetic biological system based on Escherichia coli. Strains carry 1 of 10 different plasmids and one of three genomically integrated fluorescent markers. All strains are co-cultured to obtain real-time measurements of optical density (total population density) and fluorescence (sub-population densities). We identify challenges in calibrating between fluorescence and density and of fluorescent proteins maturing at different rates. To resolve these issues, we compare two methods of fluorescence calibration and correct for maturation by measuring in vivo maturation times. We provide evidence of genetic interactions occurring between our SGCs and further show how to use our statistical model to test some hypotheses about microbial growth and the costs of protein expression.
IMPORTANCE
Fluorescently labeled co-cultures are becoming increasingly popular. The approach proposed here offers a high standard for experimental design and data analysis to measure selection coefficients and growth rates in competition. Measuring competitive differences is useful in many laboratory studies, allowing for fitness cost-correction of growth rates and ecological interactions and testing hypotheses in synthetic biology. Using time-resolved growth curves, rather than endpoint measurements, for competition assays allows us to construct a detailed scientific model that can be used to ask questions about fine-grained phenomena, such as bacterial growth dynamics, as well as higher-level phenomena, such as the interactions between synthetic cassette expression.
... Analyses of species coexistence focusing on frequency dependence also make this zero sum assumption (Godwin et al. 2020). This zero sum assumption is only valid for some models, notably Lotka-Volterra competition among similar species (Mallet 2012, Lion 2018), but it does not typically emerge from models of other interaction types DeAngelis 2009, 2010). ...
Biodiversity describes the variety of organisms on planet earth. Ecologists have long hoped for a synthesis between analyses of biodiversity and analyses of biotic interactions among species, such as predation, competition, and mutualism. However, it is often unclear how to connect details of these interactions with complex modern analyses of biodiversity. Using methods pioneered in studies of ecological-evolutionary dynamics, we link biotic interactions and changes in measures of biodiversity such as Hill numbers. We show that analyses of biodiversity obscure details about biotic interactions. For example, identical changes in biodiversity can arise from predation, competition or mutualism, locally or across a metacommunity. Our approach indicates that traditional models of community assembly miss key facets of diversity change. Instead, we suggest that analyses of diversity change should focus on partitions, which measure mechanisms that directly shape changes in diversity, notably relative fitness and immigration, rather than traditional analyses of biotic interactions.
... Some strains and media exhibit biphasic/diauxic growth (not further detailed herein), where population growth decelerates or plateaus and then accelerates again. Even in the absence of biphasic growth, the trend of the per capita growth rate as it approaches the stationary phase can depend on changes in the media environment (e.g., depletion of resources or accumulation of waste products) and/or on some types of intraspecific density-dependent effects (i.e., Allee effects) caused by cell interactions (e.g., facilitation, competition/interference, or quorum sensing; but see Mallet, 2012). For optical density data, the shape of the growth curve is usually impacted by changes in average cell size (Stevenson et al., 2016). ...
Introduction
After more than 100 years of generating monoculture batch culture growth curves, microbial ecologists and evolutionary biologists still lack a reference method for inferring growth rates. Our work highlights the challenges of estimating the growth rate from growth curve data. It shows that inaccurate estimates of growth rates significantly impact the estimated relative fitness, a principal quantity in evolution and ecology.
Methods and results
First, we conducted a literature review and found which methods are currently used to estimate growth rates. These methods differ in the meaning of the estimated growth rate parameter. Mechanistic models estimate the intrinsic growth rate µ, whereas phenomenological methods – both model-based and model-free – estimate the maximum per capita growth rate µ max. Using math and simulations, we show the conditions in which µ max is not a good estimator of µ. Then, we demonstrate that inaccurate absolute estimates of µ are not overcome by calculating relative values. Importantly, we find that poor approximations for µ sometimes lead to wrongly classifying a beneficial mutant as deleterious. Finally, we re-analyzed four published data sets, using most of the methods found in our literature review. We detected no single best-fitting model across all experiments within a data set and found that the Gompertz models, which were among the most commonly used, were often among the worst-fitting.
Discussion
Our study suggests how experimenters can improve their growth rate and associated relative fitness estimates and highlights a neglected but fundamental problem for nearly everyone who studies microbial populations in the lab.
... We study a niche-based model of competition based on generalized Lotka-Volterra dynamics following the α − r parametrization, whose advantages have been discussed by Mallet [43]. Each phenotype is defined by its position μ i on a niche axis representing resources. ...
The realization that evolutionary feedbacks need to be considered to fully grasp ecological dynamics has sparked interest in the effect of evolution on community properties like coexistence and productivity. However, little is known about the evolution of community robustness and productivity along diversification processes in species-rich systems. We leverage the recent structural approach to coexistence together with adaptive dynamics to study such properties and their relationships in a general trait-based model of competition on a niche axis. We show that the effects of coevolution on coexistence are two-fold and contrasting depending on the time scale considered. In the short term, evolution of niche differentiation strengthens coexistence, while long-term diversification leads to niche packing and decreased robustness. Moreover, we find that coevolved communities tend to be on average more robust and more productive than non-evolutionary assemblages. We illustrate how our theoretical predictions echo in observed empirical patterns and the implications of our results for empiricists and applied ecologists. We suggest that some of our results such as the improved robustness of Evolutionarily Stable Communities could be tested experimentally in suitable model systems.
... We consider Set 1 of the model parameter values presented in Table 1. In the nonspatial case, in the absence of phytophage, i.e., with Z 0 = 0, it corresponds to classical bistability of two axial equilibria in a two-species Lotka-Volterra competition model [26,27], with either weed or cultivated plant reaching its carrying capacity: R = r R /c R , P = Z = 0, and P = r P /c P , R = Z = 0. However, adding sufficient quantity of the phytophagous insects into the system alternates the model dynamics. ...
... Note that, in the absence of phytophage population and with homogeneous distribution of weed and cultivated plant, the dynamics of the System (1)-(4) in a homogeneous habitat with the Neumann boundary condition (6) obey the classical nonspatial Lotka-Volterra model of two species competition [26,27]: ...
We model the spatiotemporal dynamics of a community consisting of competing weed and cultivated plant species and a population of specialized phytophagous insects used as the weed biocontrol agent. The model is formulated as a PDE system of taxis–diffusion–reaction type and computer-implemented for one-dimensional and two-dimensional cases of spatial habitat for the Neumann zero-flux boundary condition. In order to discretize the original continuous system, we applied the method of lines. The obtained system of ODEs is integrated using the Runge–Kutta method with a variable time step and control of the integration accuracy. The numerical simulations provide insights into the mechanism of formation of solitary population waves (SPWs) of the phytophage, revealing the factors that determine the efficacy of combined application of the phytophagous insect (classical biological method) and cultivated plant (phytocenotic method) to suppress weed foci. In particular, the presented results illustrate the stabilizing action of cultivated plants, which fix the SPW effect by occupying the free area behind the wave front so that the weed remains suppressed in the absence of a phytophage.
... Despite its widespread use, it is important to recall that the logistic model is an abstract description of population dynamics (Herrando-Pérez et al. 2012). This level of abstraction makes the interpretation of parameters challenging and may lead to paradoxical behaviours (Ginzburg 1992, Gabriel et al. 2005, Mallet 2012). Such issues are especially apparent in the often used r − K formulation with r 0 being the intrinsic rate of increase, K the carrying capacity and N the population density: ...
... Recently, Reding-Roman et al. (2017) found positive r -K relationships in microbial systems counter to their initial hypothesis, which led the authors to postulate 'trade-ups' and 'uberbugs' while discussing the relevance of these findings for cancer (Aktipis et al. 2013) and antibiotic resistance research (paper highlighted by Reznick and King 2017). Although these and related issues have been discussed in detail by Matessi and Gatto (1984), Reznick et al. (2002), Rueffler et al. (2006) and Mallet (2012), to name but a few, current empirical work continues to expect negative r -K relationships Altermatt 2015, Reding-Roman et al. 2017) and some theory continues to use 'K' as an evolving trait (Lande et al. 2009, Burton et al. 2010, Engen and Saether 2017, Fleischer et al. 2018. ...
... In order to resolve some of the issues associated with the logistic growth model as described by Eq. 1, Mallet (2012), for instance, has promoted the use of Verhulst's original rα formulation of logistic growth (Kostitzin 1937, Verhulst 1838. In comparison to the popular r -K formulation (Eq. ...
The logistic growth model is one of the most frequently used formalizations of density dependence affecting population growth, persistence and evolution. Ecological and evolutionary theory, and applications to understand population change over time often include this model. However, the assumptions and limitations of this popular model are often not well appreciated. Here, we briefly review past use of the logistic growth model and highlight limitations by deriving population growth models from underlying consumer–resource dynamics. We show that the logistic equation likely is not applicable to many biological systems. Rather, density‐regulation functions are usually non‐linear and may exhibit convex or concave curvatures depending on the biology of resources and consumers. In simple cases, the dynamics can be fully described by the Schoener model. More complex consumer dynamics show similarities to a Maynard Smith–Slatkin model. We show how population‐level parameters, such as intrinsic rates of increase and equilibrium population densities are not independent, as often assumed. Rather, they are functions of the same underlying parameters. The commonly assumed positive relationship between equilibrium population density and competitive ability is typically invalid. We propose simple relationships between intrinsic rates of increase and equilibrium population densities that capture the essence of different consumer–resource systems. Relating population level models to underlying mechanisms allows us to discuss applications to evolutionary outcomes and how these models depend on environmental conditions, like temperature via metabolic scaling. Finally, we use time‐series from microbial food chains to fit population growth models as a test case for our theoretical predictions. Our results show that density‐regulation functions need to be chosen carefully as their shapes will depend on the study system's biology. Importantly, we provide a mechanistic understanding of relationships between model parameters, which has implications for theory and for formulating biologically sound and empirically testable predictions.
... In their simplest form, the dynamics of populations are described in terms of two parameters: r, the intrinsic rate of increase; and K, the carrying capacity of the population. These two parameters are fundamental to population ecology and have a long history of empirical and theoretical study [1]. From an evolutionary perspective, r and K were used to define and describe different modes of life: r-strategists were thought to have fast population growth rates at the expense of poor competitive abilities; Kstrategists were thought to have slow-growing populations but be superior competitors, or at least more efficient with regards to resources [2]. ...
... The intrinsic rate of increase, r, and the carrying capacity of populations, K, are much more related than most people realise. Jim Mallet provides a useful history of this issue and we refer readers to his elegant exposition for more detail [1]. Briefly, the most common formulation of the logistic equation for population growth, which includes the familiar r and K, came about because one influential text published a century ago used that particular formulation, where per capita population growth rate (R) is given by Eq. (1): ...
... This equation, plus observations that species with high values of r tend to have lower values of K, led to the idea that r and K are biological parameters that show covariance but are also 'separate' from each other in a mathematical sense. However, as multiple authors have pointed out over time [1,3,10], the original formulation of the logistic equation was more focused on the biological processes rather than the outcomes of those processes, having the form of Eq. (2): ...
