Front acceleration by dynamic selection in Fisher population waves

Simulating the dynamics of dispersal and dispersal ability in fragmented populations with mate-finding Allee effects

We consider the spatial propagation and genetic evolution of model populations comprising multiple subpopulations, each distinguished by its own characteristic dispersal rate. Mate finding is modeled in accord with the assumption that reproduction is based on random encounters between pairs of individuals, so that the frequency of interbreeding between two subpopulations is proportional to the product of local population densities of each. The resulting nonlinear growth term produces an Allee effect, whereby reproduction rates are lower in sparsely populated areas; the distribution of dispersal rates that evolves is then highly dependent upon the population's initial spatial distribution. In a series of numerical test cases, we consider how these dynamics affect lattice-like arrangements of population fragments, and investigate how a population's initial fragmentation determines the dispersal rates that evolve as a habitat is colonized. First, we consider a case where initial population fragments coincide with habitat islands, within which death rates differ from those that apply outside; the presence of inhospitable exterior regions exaggerates Allee effect-driven reductions in dispersal ability. We then examine how greater distances separating adjacent population fragments lead to more severe reductions in dispersal ability. For populations of a fixed initial magnitude, fragmentation into smaller, denser patches leads not only to greater losses of dispersal ability, but also helps ensure the population's long-term persistence, emphasizing the trade-offs between the benefits and risks of rapid dispersal under Allee effects. Next, simulations of well-established populations disrupted by localized depopulation events illustrate how mate-finding Allee effects and spatial heterogeneity can drive a population's dispersal ability to evolve either downward or upward depending on conditions, highlighting a qualitative distinction between population fragmentation and habitat heterogeneity. A final test case compares populations that are fragmented across multiple scales, demonstrating how differences in the relative scales of micro- and macro-level fragmentation can lead to qualitatively different evolutionary outcomes.

Ancestral Lineages in Mutation Selection Equilibria with Moving Optimum

Many populations can somehow adapt to rapid environmental changes. To understand this fast evolution, we investigate the genealogy of individuals inside those populations. More precisely, we use a deterministic model to describe the phenotypic density of a population under selection when the fitness optimum moves at constant speed. We study the inside dynamics of this population using the neutral fractions approach. We then define a Markov process characterizing the distribution of ancestral phenotypic lineages inside the equilibrium. This construction yields qualitative as well as quantitative properties on the phenotype of typical ancestors. In particular, we show that in asexual populations typical ancestors of present individuals carried traits much closer to the fitness optimum than most individuals alive at the same time. We also investigate more deeply the asymptotic regime of small mutation effects. In this regime, we obtain an explicit formula for the typical ancestral lineage using the description of the solutions of the Hamilton Jacobi equation as a minimizer of an optimization problem. In addition, we compare our deterministic results on lineages with the lineages of stochastic models.

Spatial dynamics of a nonlocal predator–prey model with double mutation

In this paper, we are dedicated to studying the global dynamics of a nonlocal predator–prey model with double mutation. First, by defining a pair of upper and lower solutions, we build a new comparison principle. Furthermore, based on the new comparison principle, we get the existence of the solutions by constructing monotone iterative sequences. Finally, using the quasi-fundamental solution, Gronwall’s inequality and auxiliary functions, the uniqueness and uniform boundedness of the solutions are given. It is worth noting that this paper is the first to introduce the double mutation into the system and obtain the well-posedness and the uniform boundedness of the solutions by more detailed analysis and estimation.

Slow expanders invade by forming dented fronts in microbial colonies

Living organisms never cease to evolve, so there is a significant interest in predicting and controlling evolution in all branches of life sciences. The most basic question is whether a trait should increase or decrease in a given environment. The answer seems to be trivial for traits such as the growth rate in a bioreactor or the expansion rate of a tumor. Yet, it has been suggested that such traits can decrease, rather than increase, during evolution. Here, we report a mutant that outcompeted the ancestor despite having a slower expansion velocity when in isolation. To explain this observation, we developed and validated a theory that describes spatial competition between organisms with different expansion rates and arbitrary competitive interactions.

Most organisms grow in space, whether they are viruses spreading within a host tissue or invasive species colonizing a new continent. Evolution typically selects for higher expansion rates during spatial growth, but it has been suggested that slower expanders can take over under certain conditions. Here, we report an experimental observation of such population dynamics. We demonstrate that mutants that grow slower in isolation nevertheless win in competition, not only when the two types are intermixed, but also when they are spatially segregated into sectors. The latter was thought to be impossible because previous studies focused exclusively on the global competitions mediated by expansion velocities, but overlooked the local competitions at sector boundaries. Local competition, however, can enhance the velocity of either type at the sector boundary and thus alter expansion dynamics. We developed a theory that accounts for both local and global competitions and describes all possible sector shapes. In particular, the theory predicted that a slower on its own, but more competitive, mutant forms a dented V-shaped sector as it takes over the expansion front. Such sectors were indeed observed experimentally, and their shapes matched quantitatively with the theory. In simulations, we further explored several mechanisms that could provide slow expanders with a local competitive advantage and showed that they are all well-described by our theory. Taken together, our results shed light on previously unexplored outcomes of spatial competition and establish a universal framework to understand evolutionary and ecological dynamics in expanding populations.

Superlinear growth reveals the Allee effect in tumors

Integrating experimental data into ecological models plays a central role in understanding biological mechanisms that drive tumor progression where such knowledge can be used to develop new therapeutic strategies. While the current studies emphasize the role of competition among tumor cells, they fail to explain recently observed superlinear growth dynamics across human tumors. Here we study tumor growth dynamics by developing a model that incorporates evolutionary dynamics inside tumors with tumor-microenvironment interactions. Our results reveal that tumor cells' ability to manipulate the environment and induce angiogenesis drives superlinear growth-a process compatible with the Allee effect. In light of this understanding, our model suggests that, for high-risk tumors that have a higher growth rate, suppressing angiogenesis can be the appropriate therapeutic intervention.

Mix and Match: Phenotypic Coexistence as a Key Facilitator of Cancer Invasion

Invasion of healthy tissue is a defining feature of malignant tumours. Traditionally, invasion is thought to be driven by cells that have acquired all the necessary traits to overcome the range of biological and physical defences employed by the body. However, in light of the ever-increasing evidence for geno- and phenotypic intra-tumour heterogeneity, an alternative hypothesis presents itself: could invasion be driven by a collection of cells with distinct traits that together facilitate the invasion process? In this paper, we use a mathematical model to assess the feasibility of this hypothesis in the context of acid-mediated invasion. We assume tumour expansion is obstructed by stroma which inhibits growth and extra-cellular matrix (ECM) which blocks cancer cell movement. Further, we assume that there are two types of cancer cells: (i) a glycolytic phenotype which produces acid that kills stromal cells and (ii) a matrix-degrading phenotype that locally remodels the ECM. We extend the Gatenby-Gawlinski reaction-diffusion model to derive a system of five coupled reaction-diffusion equations to describe the resulting invasion process. We characterise the spatially homogeneous steady states and carry out a simulation study in one spatial dimension to determine how the tumour develops as we vary the strength of competition between the two phenotypes. We find that overall tumour growth is most extensive when both cell types can stably coexist, since this allows the cells to locally mix and benefit most from the combination of traits. In contrast, when inter-species competition exceeds intra-species competition the populations spatially separate and invasion arrests either: (i) rapidly (matrix-degraders dominate) or (ii) slowly (acid-producers dominate). Overall, our work demonstrates that the spatial and ecological relationship between a heterogeneous population of tumour cells is a key factor in determining their ability to cooperate. Specifically, we predict that tumours in which different phenotypes coexist stably are more invasive than tumours in which phenotypes are spatially separated.

Evolution at the Edge of Expanding Populations

Predicting the evolution of expanding populations is critical to controlling biological threats such as invasive species and cancer metastasis. Expansion is primarily driven by reproduction and dispersal, but nature abounds with examples of evolution where organisms pay a reproductive cost to disperse faster. When does selection favor this "survival of the fastest"? We searched for a simple rule, motivated by evolution experiments where swarming bacteria evolved into a hyperswarmer mutant that disperses ∼100% faster but pays a growth cost of ∼10% to make many copies of its flagellum. We analyzed a two-species model based on the Fisher equation to explain this observation: the population expansion rate (v) results from an interplay of growth (r) and dispersal (D) and is independent of the carrying capacity: v = 2 ( rD ) 1 / 2 . A mutant can take over the edge only if its expansion rate (v2) exceeds the expansion rate of the established species (v1); this simple condition ( v 2 > v 1 ) determines the maximum cost in slower growth that a faster mutant can pay and still be able to take over. Numerical simulations and time-course experiments where we tracked evolution by imaging bacteria suggest that our findings are general: less favorable conditions delay but do not entirely prevent the success of the fastest. Thus, the expansion rate defines a traveling wave fitness, which could be combined with trade-offs to predict evolution of expanding populations.

Local adaptation, dispersal evolution, and the spatial eco-evolutionary dynamics of invasion

Local adaptation and dispersal evolution are key evolutionary processes shaping the invasion dynamics of populations colonizing new environments. Yet their interaction is largely unresolved. Using a single-species population model along a one-dimensional environmental gradient, we show how local competition and dispersal jointly shape the eco-evolutionary dynamics and speed of invasion. From a focal introduction site, the generic pattern predicted by our model features a temporal transition from wave-like to pulsed invasion. Each regime is driven primarily by local adaptation, while the transition is caused by eco-evolutionary feedbacks mediated by dispersal. The interaction range and cost of dispersal arise as key factors of the duration and speed of each phase. Our results demonstrate that spatial eco-evolutionary feedbacks along environmental gradients can drive strong temporal variation in the rate and structure of population spread, and must be considered to better understand and forecast invasion rates and range dynamics.

Kinetic Modeling of Local Epidemic Spread and Its Simulation

The local epidemic spread in physical space is modeled using the kinetic equation. In particular, the infection occurs via the binary interaction between the uninfected and infected individuals. Then, the local epidemic spread can be modeled on the basis of the stochastic Boltzmann type equation. In this paper, the normalized virus titer inside the infected human body is defined as the function of the elapsed time, which is measured from the infection time. Consequently, the probability of the infection at the binary human interaction increases, as the normalized virus titer inside the human body increases, whereas the normalized virus titer inside the infected human body decreases, after the normalized virus titer reaches to its maximum value, namely, unity, in the characteristic time. Numerical results indicate that the propagation speed of the boundary between the infected and uninfected domains depends on such a characteristic time, strongly, when the Knudsen number and temperature are fixed. Such a dependency of the propagation speed of the boundary between the infected and uninfected domains on the characteristic time can be described by the Fisher-Kolmogorov-Petrovsky-Piscounov equation which is introduced from the stochastic Boltzmann type equation. Finally, we consider three types of the human behavior as plausible actions to the local epidemic spread.

After the games are over: life-history trade-offs drive dispersal attenuation following range expansion

Increased dispersal propensity often evolves on expanding range edges due to the Olympic Village effect, which involves the fastest and fittest finding themselves together in the same place at the same time, mating, and giving rise to like individuals. But what happens after the range's leading edge has passed and the games are over? Although empirical studies indicate that dispersal propensity attenuates following range expansion, hypotheses about the mechanisms driving this attenuation have not been clearly articulated or tested. Here, we used a simple model of the spatiotemporal dynamics of two phenotypes, one fast and the other slow, to propose that dispersal attenuation beyond preexpansion levels is only possible in the presence of trade‐offs between dispersal and life‐history traits. The Olympic Village effect ensures that fast dispersers preempt locations far from the range's previous limits. When trade‐offs are absent, this preemptive spatial advantage has a lasting impact, with highly dispersive individuals attaining equilibrium frequencies that are strictly higher than their introduction frequencies. When trade‐offs are present, dispersal propensity decays rapidly at all locations. Our model's results about the postcolonization trajectory of dispersal evolution are clear and, in principle, should be observable in field studies. We conclude that empirical observations of postcolonization dispersal attenuation offer a novel way to detect the existence of otherwise elusive trade‐offs between dispersal and life‐history traits.

Evolution Arrests Invasions of Cooperative Populations

Population expansions trigger many biomedical and ecological transitions, from tumor growth to invasions of non-native species. Although population spreading often selects for more invasive phenotypes, we show that this outcome is far from inevitable. In cooperative populations, mutations reducing dispersal have a competitive advantage. Such mutations then steadily accumulate at the expansion front, bringing invasion to a halt. Our findings are a rare example of evolution driving the population into an unfavorable state, and they could lead to new strategies to combat unwelcome invaders.

Spatial assortment of mixed propagules explains the acceleration of range expansion

Range expansion of spreading organisms has been found to follow three types: (i) linear expansion with a constant rate of spread; (ii) bi-phase expansion with a faster linear expansion following a slower linear expansion; and (iii) accelerating expansion with a continuously increasing rate of spread. To date, no overarching formula exists that can be applied to all three types of range expansion. We investigated how propagule pressure, i.e., the initial number of individuals and their composition in terms of dispersal ability, affects the spread of a population. A system of integrodifference equations was then used to model the spatiotemporal dynamics of the population. We studied the dynamics of dispersal ability as well as the instantaneous and asymptotic rate of spread. We found that individuals with different dispersal abilities were spatially sorted with the stronger dispersers situated at the expanding range front, causing the velocity of expansion to accelerate. The instantaneous rate of spread was found to be fully determined by the growth and dispersal abilities of the population at the advancing edge of the invasion. We derived a formula for the asymptotic rate of spread under different scenarios of propagule pressure. The results suggest that data collected from the core of the invasion may underestimate the spreading rate of the population. Aside from better managing of invasive species, the derived formula could conceivably also be applied to conservation management of relocated, endangered or extra-limital species.

Selective advantage of diffusing faster

We study a stochastic spatial model of biological competition in which two species have the same birth and death rates, but different diffusion constants. In the absence of this difference, the model can be considered as an off-lattice version of the voter model and presents similar coarsening properties. We show that even a relative difference in diffusivity on the order of a few percent may lead to a strong bias in the coarsening process favoring the more agile species. We theoretically quantify this selective advantage and present analytical formulas for the average growth of the fastest species and its fixation probability.

Range expansion of heterogeneous populations

Risk spreading in bacterial populations is generally regarded as a strategy to maximize survival. Here, we study its role during range expansion of a genetically diverse population where growth and motility are two alternative traits. We find that during the initial expansion phase fast-growing cells do have a selective advantage. By contrast, asymptotically, generalists balancing motility and reproduction are evolutionarily most successful. These findings are rationalized by a set of coupled Fisher equations complemented by stochastic simulations.

Rapid increase in dispersal during range expansion in the invasive ladybird Harmonia axyridis

The evolutionary trajectories associated with demographic, genetic and spatial disequilibrium have become an issue of growing interest in population biology. Invasive species provide unique opportunities to explore the impact of recent range expansion on life-history traits, making it possible to test for a spatial arrangement of dispersal abilities along the expanding range, in particular. We carried out controlled experiments in laboratory conditions to test the hypothesis of an increase in dispersal capacity with range expansion in Harmonia axyridis, a ladybird that has been invading Europe since 2001. We found a marked increase in the flight speed of the insects from the core to the front of the invasion range in two independent sampling transects. By contrast, we found that two other traits associated with dispersal (endurance and motivation to fly off) did not follow the same spatial gradient. Our results provide a striking illustration of the way in which predictable directional genetic changes may occur rapidly for some traits associated with dispersal during biological invasions. We discuss the consequences of our results for invasion dynamics and the evolutionary outcomes of spatially expanding populations.

Evolution of dispersal and life history interact to drive accelerating spread of an invasive species

Populations on the edge of an expanding range are subject to unique evolutionary pressures acting on their life-history and dispersal traits. Empirical evidence and theory suggest that traits there can evolve rapidly enough to interact with ecological dynamics, potentially giving rise to accelerating spread. Nevertheless, which of several evolutionary mechanisms drive this interaction between evolution and spread remains an open question. We propose an integrated theoretical framework for partitioning the contributions of different evolutionary mechanisms to accelerating spread, and we apply this model to invasive cane toads in northern Australia. In doing so, we identify a previously unrecognised evolutionary process that involves an interaction between life-history and dispersal evolution during range shift. In roughly equal parts, life-history evolution, dispersal evolution and their interaction led to a doubling of distance spread by cane toads in our model, highlighting the potential importance of multiple evolutionary processes in the dynamics of range expansion.

Tumor evolution in space: the effects of competition colonization tradeoffs on tumor invasion dynamics

We apply competition colonization tradeoff models to tumor growth and invasion dynamics to explore the hypothesis that varying selection forces will result in predictable phenotypic differences in cells at the tumor invasive front compared to those in the core. Spatially, ecologically, and evolutionarily explicit partial differential equation models of tumor growth confirm that spatial invasion produces selection pressure for motile phenotypes. The effects of the invasive phenotype on normal adjacent tissue determine the patterns of growth and phenotype distribution. If tumor cells do not destroy their environment, colonizer and competitive phenotypes coexist with the former localized at the invasion front and the latter, to the tumor interior. If tumors cells do destroy their environment, then cell motility is strongly selected resulting in accelerated invasion speed with time. Our results suggest that the widely observed genetic heterogeneity within cancers may not be the stochastic effect of random mutations. Rather, it may be the consequence of predictable variations in environmental selection forces and corresponding phenotypic adaptations.