We present a theoretical model of evolution of spatially distributed populations in which organis... more We present a theoretical model of evolution of spatially distributed populations in which organisms mate with and compete against each other only locally. We show using both analysis and numerical simulation that the typical dynamics of population density variation is a spontaneous formation of isolated groups due to competition for resources. The resulting spatial separation between groups strongly affects the process of genetic invasion by local reproductive mixing, and spatially inhomogeneous genetic distributions are possible in the final states. We then consider a specific version of this model in the presence of disruptive selection, favoring two fittest types against their genetic intermediates. This case can be simplified to a system that involves just two nonconserved order parameters: population density and type difference. Since the coexistence of two fittest types is unstable in this case, symmetry breaking and coarsening occur in type difference, implying eventual dominance by one type over another for finite populations. However, such coarsening patterns may be pinned by the spontaneously generated spatial separation between isolated groups. The longterm evolution of genetic composition is found to be sensitive to the ratio of the mating and competition ranges, and other parameters. Our model may provide a theoretical basis for consideration of various properties of spatially extended evolutionary processes, including spontaneous formation of subpopulations and lateral invasion of different types.
Mutualistic networks are crucial to the maintenance of ecosystem services. Unfortunately, what we... more Mutualistic networks are crucial to the maintenance of ecosystem services. Unfortunately, what we know about seed dispersal networks is based only on bird-fruit interactions. Therefore, we aimed at filling part of this gap by investigating bat-fruit networks. It is known from population studies that: (i) some bat species depend more on fruits than others, and (ii) that some specialized frugivorous bats prefer particular plant genera. We tested whether those preferences affected the structure and robustness of the whole network and the functional roles of species. Nine bat-fruit datasets from the literature were analyzed and all networks showed lower complementary specialization (H 2 ' = 0.3760.10, mean 6 SD) and similar nestedness (NODF = 0.5660.12) than pollination networks. All networks were modular (M = 0.3260.07), and had on average four cohesive subgroups (modules) of tightly connected bats and plants. The composition of those modules followed the genus-genus associations observed at population level (Artibeus-Ficus, Carollia-Piper, and Sturnira-Solanum), although a few of those plant genera were dispersed also by other bats. Bat-fruit networks showed high robustness to simulated cumulative removals of both bats (R = 0.5560.10) and plants (R = 0.6860.09). Primary frugivores interacted with a larger proportion of the plants available and also occupied more central positions; furthermore, their extinction caused larger changes in network structure. We conclude that bat-fruit networks are highly cohesive and robust mutualistic systems, in which redundancy is high within modules, although modules are complementary to each other. Dietary specialization seems to be an important structuring factor that affects the topology, the guild structure and functional roles in bat-fruit networks.
We compute the semiclassical coherent-state propagator for a particle moving in a one-dimensional... more We compute the semiclassical coherent-state propagator for a particle moving in a one-dimensional box. In this semiclassical approach complex trajectories are stationary paths of the propagator's asymptotic expansion and play a fundamental role. A second semiclassical approximation is also introduced, which makes use of real trajectories only. An application to a seemingly simple system, the infinite well, is carried out completely for the diagonal elements, and a comparison is made among the three possible methods, those based on complex and real trajectories and the ''exact case'' that is determined by decomposing the propagator into its eigenstates.
The Kirchhoff model describes the statics and dynamics of thin rods within the approximations of ... more The Kirchhoff model describes the statics and dynamics of thin rods within the approximations of the linear elasticity theory. In this paper we develop a method, based on a shooting technique, to find equilibrium configurations of finite rods subjected to boundary conditions and given load parameters. The method consists in making a series of small changes on a trial solution satisfying the Kirchhoff equations but not necessarily the boundary conditions. By linearizing the differential equations around the trial solution we are able to push its end point to the desired position, step by step. The method is also useful to obtain configurations of rods with fixed end points but different mechanical parameters, such as tension, components of the moment or inhomogeneities.
Physica A: Statistical Mechanics and its Applications, 2010
Predators and preys often form species networks with asymmetric patterns of interaction. We study... more Predators and preys often form species networks with asymmetric patterns of interaction. We study the dynamics of a four species network consisting of two weakly connected predator-prey pairs. We focus our analysis on the effects of the cross interaction between the predator of the first pair and the prey of the second pair. This is an example where the predator overlap, which is the proportion of predators that a given prey shares with other preys, is not uniform across the network due to asymmetries in patterns of interaction. We explore the behavior of the system under different interaction strengths and study the dynamics of survival and extinction. In particular, we consider situations in which the four species have initial populations lower than their long-term equilibrium, simulating catastrophic situations in which their abundances are reduced due to human action or environmental change. We show that, under these reduced initial conditions, and depending on the strength of the cross interaction, the populations tend to oscillate before re-equilibrating, disturbing the community equilibrium and sometimes reaching values that are only a small fraction of the equilibrium population, potentially leading to their extinction. We predict that, contrary to one's intuition, the most likely scenario is the extinction of the less predated preys.
In networks of plant-animal mutualisms, different animal groups interact preferentially with diff... more In networks of plant-animal mutualisms, different animal groups interact preferentially with different plants, thus forming distinct modules responsible for different parts of the service. However, what we currently know about seed dispersal networks is based only on birds. Therefore, we wished to fill this gap by studying bat-fruit networks and testing how they differ from bird-fruit networks. As dietary overlap of Neotropical bats and birds is low, they should form distinct mutualistic modules within local networks. Furthermore, since frugivory evolved only once among Neotropical bats, but several times independently among Neotropical birds, greater dietary overlap is expected among bats, and thus connectance and nestedness should be higher in bat-fruit networks. If bat-fruit networks have higher nestedness and connectance, they should be more robust to extinctions. We analyzed 1 mixed network of both bats and birds and 20 networks that consisted exclusively of either bats (11) or birds (9). As expected, the structure of the mixed network was both modular (M = 0.45) and nested (NODF = 0.31); one module contained only birds and two only bats. In 20 datasets with only one disperser group, bat-fruit networks (NODF = 0.53 ± 0.09, C = 0.30 ± 0.11) were more nested and had a higher connectance than bird-fruit networks (NODF = 0.42 ± 0.07, C = 0.22 ± 0.09). Unexpectedly, robustness to extinction of animal species was higher in bird-fruit networks (R = 0.60 ± 0.13) than in bat-fruit networks (R = 0.54 ± 0.09), and differences were explained mainly by species richness. These findings suggest that a modular structure also occurs in seed dispersal networks, similar to pollination networks. The higher nestedness and connectance observed in bat-fruit networks compared with bird-fruit networks may be explained by the monophyletic evolution of frugivory in Neotropical bats, among which the diets of specialists seem to have evolved from the pool of fruits consumed by generalists.
The frequency distribution of the number of interactions per species (i.e., degree distribution) ... more The frequency distribution of the number of interactions per species (i.e., degree distribution) within plant-animal mutualistic assemblages often decays as a power-law with an exponential truncation. Such a truncation suggests that there are ecological factors limiting the frequency of supergeneralist species. However, it is not clear whether these patterns can emerge from intrinsic features of the interacting assemblages, such as differences between plant and animal species richness (richness ratio). Here, we show that high richness ratios often characterize plant-animal mutualisms. Then, we demonstrate that exponential truncations are expected in bipartite networks generated by a simple model that incorporates build-up mechanisms that lead to a high richness ratio. Our results provide a simple interpretation for the truncations commonly observed in the degree distributions of mutualistic networks that complements previous ones based on biological effects. r
Many quantitative genetic and adaptive dynamic models suggest that disruptive selection can maint... more Many quantitative genetic and adaptive dynamic models suggest that disruptive selection can maintain genetic polymorphism and be the driving force causing evolutionary divergence. These models also suggest that disruptive selection arises from frequency-dependent intraspecific competition. For convenience or historical precedence, these models assume that carrying capacity and competition functions follow a Gaussian distribution. Here, we propose a new analytical framework that relaxes the assumption of Gaussian competition and carrying capacity functions, and investigate how alternative shapes affect the likelihood of disruptive selection. We found that the shape of both carrying capacity and competition kernels interact to determine the likelihood of disruptive selection. For certain regions of the parametric space disruptive selection is facilitated, whereas for others it becomes more difficult. Our results suggest that the relationship between the degree of frequency dependence and the likelihood of disruptive selection is more complex than previously thought, depending on how resources are distributed and competition interference takes place. It is now important to describe the empirical patterns of resource distribution and competition in nature as a way to determine the likelihood of disruptive selection in natural populations.
The notion of modularity has become a unifying principle to understand structural and functional ... more The notion of modularity has become a unifying principle to understand structural and functional aspects of biological organization at different levels of complexity. Recently, deciphering the modular organization of molecular systems has been greatly aided by network theory. Nevertheless, network theory is completely absent from the investigation of modularity of complex macroscopic phenotypes, a fundamental level of organization at which organisms experience and interact with the environment. Here, we used geometric descriptors of phenotypic variation to derive a network representation of a complex morphological structure, the mammalian mandible, in terms of nodes and links. Then, by integrating the network representation and description with random matrix theory, we uncovered a modular organization for the mammalian mandible, which deviates significantly from an equivalent random network. The modules revealed by the network analysis correspond to the four morphogenetic units recognized for the mammalian mandible on a developmental basis. Furthermore, these modules are known to be affected only by particular genes and are also functionally differentiated. This study shows that the powerful formalism of network theory can be applied to the discovery of modules in complex phenotypes and opens the possibility of an integrated approach to the study of modularity at all levels of organizational complexity.
The model of Hastings and Powell describes a tritrophic food chain that exhibits chaotic dynamics... more The model of Hastings and Powell describes a tritrophic food chain that exhibits chaotic dynamics. The model assumes that the populations are homogeneously mixed, so that the probability that any two individuals interact is uniform and space can be ignored. In this paper we propose a spatial version of the Hastings and Powell model in which predators seek their preys only in a finite neighborhood of their home location, breaking the mixing hypothesis. Treating both space and time as discrete variables we derive a set of coupled equations that describe the evolution of the populations at each site of the spatial domain. We show that the introduction of local predator-prey interactions result in qualitatively distinct dynamics of predator and prey populations. The evolution equations for the predators involve averages over the local density of preys, whereas the equations for the preys involve double averages, where the local density of both preys and predators appear. Our numerical simulations show that local predation also leads to spontaneous pattern formation and to qualitative changes in the global dynamics of the system. In particular, depending on the size of the predation neighborhoods, the chaotic strange attractor present in the original model of Hastings and Powell can be replaced by a stable fixed point or by an attractor of simpler topology. (M.A.M. de Aguiar).
We present a theoretical model of evolution of spatially distributed populations in which organis... more We present a theoretical model of evolution of spatially distributed populations in which organisms mate with and compete against each other only locally. We show using both analysis and numerical simulation that the typical dynamics of population density variation is a spontaneous formation of isolated groups due to competition for resources. The resulting spatial separation between groups strongly affects the process of genetic invasion by local reproductive mixing, and spatially inhomogeneous genetic distributions are possible in the final states. We then consider a specific version of this model in the presence of disruptive selection, favoring two fittest types against their genetic intermediates. This case can be simplified to a system that involves just two nonconserved order parameters: population density and type difference. Since the coexistence of two fittest types is unstable in this case, symmetry breaking and coarsening occur in type difference, implying eventual dominance by one type over another for finite populations. However, such coarsening patterns may be pinned by the spontaneously generated spatial separation between isolated groups. The longterm evolution of genetic composition is found to be sensitive to the ratio of the mating and competition ranges, and other parameters. Our model may provide a theoretical basis for consideration of various properties of spatially extended evolutionary processes, including spontaneous formation of subpopulations and lateral invasion of different types.
Mutualistic networks are crucial to the maintenance of ecosystem services. Unfortunately, what we... more Mutualistic networks are crucial to the maintenance of ecosystem services. Unfortunately, what we know about seed dispersal networks is based only on bird-fruit interactions. Therefore, we aimed at filling part of this gap by investigating bat-fruit networks. It is known from population studies that: (i) some bat species depend more on fruits than others, and (ii) that some specialized frugivorous bats prefer particular plant genera. We tested whether those preferences affected the structure and robustness of the whole network and the functional roles of species. Nine bat-fruit datasets from the literature were analyzed and all networks showed lower complementary specialization (H 2 ' = 0.3760.10, mean 6 SD) and similar nestedness (NODF = 0.5660.12) than pollination networks. All networks were modular (M = 0.3260.07), and had on average four cohesive subgroups (modules) of tightly connected bats and plants. The composition of those modules followed the genus-genus associations observed at population level (Artibeus-Ficus, Carollia-Piper, and Sturnira-Solanum), although a few of those plant genera were dispersed also by other bats. Bat-fruit networks showed high robustness to simulated cumulative removals of both bats (R = 0.5560.10) and plants (R = 0.6860.09). Primary frugivores interacted with a larger proportion of the plants available and also occupied more central positions; furthermore, their extinction caused larger changes in network structure. We conclude that bat-fruit networks are highly cohesive and robust mutualistic systems, in which redundancy is high within modules, although modules are complementary to each other. Dietary specialization seems to be an important structuring factor that affects the topology, the guild structure and functional roles in bat-fruit networks.
We compute the semiclassical coherent-state propagator for a particle moving in a one-dimensional... more We compute the semiclassical coherent-state propagator for a particle moving in a one-dimensional box. In this semiclassical approach complex trajectories are stationary paths of the propagator's asymptotic expansion and play a fundamental role. A second semiclassical approximation is also introduced, which makes use of real trajectories only. An application to a seemingly simple system, the infinite well, is carried out completely for the diagonal elements, and a comparison is made among the three possible methods, those based on complex and real trajectories and the ''exact case'' that is determined by decomposing the propagator into its eigenstates.
The Kirchhoff model describes the statics and dynamics of thin rods within the approximations of ... more The Kirchhoff model describes the statics and dynamics of thin rods within the approximations of the linear elasticity theory. In this paper we develop a method, based on a shooting technique, to find equilibrium configurations of finite rods subjected to boundary conditions and given load parameters. The method consists in making a series of small changes on a trial solution satisfying the Kirchhoff equations but not necessarily the boundary conditions. By linearizing the differential equations around the trial solution we are able to push its end point to the desired position, step by step. The method is also useful to obtain configurations of rods with fixed end points but different mechanical parameters, such as tension, components of the moment or inhomogeneities.
Physica A: Statistical Mechanics and its Applications, 2010
Predators and preys often form species networks with asymmetric patterns of interaction. We study... more Predators and preys often form species networks with asymmetric patterns of interaction. We study the dynamics of a four species network consisting of two weakly connected predator-prey pairs. We focus our analysis on the effects of the cross interaction between the predator of the first pair and the prey of the second pair. This is an example where the predator overlap, which is the proportion of predators that a given prey shares with other preys, is not uniform across the network due to asymmetries in patterns of interaction. We explore the behavior of the system under different interaction strengths and study the dynamics of survival and extinction. In particular, we consider situations in which the four species have initial populations lower than their long-term equilibrium, simulating catastrophic situations in which their abundances are reduced due to human action or environmental change. We show that, under these reduced initial conditions, and depending on the strength of the cross interaction, the populations tend to oscillate before re-equilibrating, disturbing the community equilibrium and sometimes reaching values that are only a small fraction of the equilibrium population, potentially leading to their extinction. We predict that, contrary to one's intuition, the most likely scenario is the extinction of the less predated preys.
In networks of plant-animal mutualisms, different animal groups interact preferentially with diff... more In networks of plant-animal mutualisms, different animal groups interact preferentially with different plants, thus forming distinct modules responsible for different parts of the service. However, what we currently know about seed dispersal networks is based only on birds. Therefore, we wished to fill this gap by studying bat-fruit networks and testing how they differ from bird-fruit networks. As dietary overlap of Neotropical bats and birds is low, they should form distinct mutualistic modules within local networks. Furthermore, since frugivory evolved only once among Neotropical bats, but several times independently among Neotropical birds, greater dietary overlap is expected among bats, and thus connectance and nestedness should be higher in bat-fruit networks. If bat-fruit networks have higher nestedness and connectance, they should be more robust to extinctions. We analyzed 1 mixed network of both bats and birds and 20 networks that consisted exclusively of either bats (11) or birds (9). As expected, the structure of the mixed network was both modular (M = 0.45) and nested (NODF = 0.31); one module contained only birds and two only bats. In 20 datasets with only one disperser group, bat-fruit networks (NODF = 0.53 ± 0.09, C = 0.30 ± 0.11) were more nested and had a higher connectance than bird-fruit networks (NODF = 0.42 ± 0.07, C = 0.22 ± 0.09). Unexpectedly, robustness to extinction of animal species was higher in bird-fruit networks (R = 0.60 ± 0.13) than in bat-fruit networks (R = 0.54 ± 0.09), and differences were explained mainly by species richness. These findings suggest that a modular structure also occurs in seed dispersal networks, similar to pollination networks. The higher nestedness and connectance observed in bat-fruit networks compared with bird-fruit networks may be explained by the monophyletic evolution of frugivory in Neotropical bats, among which the diets of specialists seem to have evolved from the pool of fruits consumed by generalists.
The frequency distribution of the number of interactions per species (i.e., degree distribution) ... more The frequency distribution of the number of interactions per species (i.e., degree distribution) within plant-animal mutualistic assemblages often decays as a power-law with an exponential truncation. Such a truncation suggests that there are ecological factors limiting the frequency of supergeneralist species. However, it is not clear whether these patterns can emerge from intrinsic features of the interacting assemblages, such as differences between plant and animal species richness (richness ratio). Here, we show that high richness ratios often characterize plant-animal mutualisms. Then, we demonstrate that exponential truncations are expected in bipartite networks generated by a simple model that incorporates build-up mechanisms that lead to a high richness ratio. Our results provide a simple interpretation for the truncations commonly observed in the degree distributions of mutualistic networks that complements previous ones based on biological effects. r
Many quantitative genetic and adaptive dynamic models suggest that disruptive selection can maint... more Many quantitative genetic and adaptive dynamic models suggest that disruptive selection can maintain genetic polymorphism and be the driving force causing evolutionary divergence. These models also suggest that disruptive selection arises from frequency-dependent intraspecific competition. For convenience or historical precedence, these models assume that carrying capacity and competition functions follow a Gaussian distribution. Here, we propose a new analytical framework that relaxes the assumption of Gaussian competition and carrying capacity functions, and investigate how alternative shapes affect the likelihood of disruptive selection. We found that the shape of both carrying capacity and competition kernels interact to determine the likelihood of disruptive selection. For certain regions of the parametric space disruptive selection is facilitated, whereas for others it becomes more difficult. Our results suggest that the relationship between the degree of frequency dependence and the likelihood of disruptive selection is more complex than previously thought, depending on how resources are distributed and competition interference takes place. It is now important to describe the empirical patterns of resource distribution and competition in nature as a way to determine the likelihood of disruptive selection in natural populations.
The notion of modularity has become a unifying principle to understand structural and functional ... more The notion of modularity has become a unifying principle to understand structural and functional aspects of biological organization at different levels of complexity. Recently, deciphering the modular organization of molecular systems has been greatly aided by network theory. Nevertheless, network theory is completely absent from the investigation of modularity of complex macroscopic phenotypes, a fundamental level of organization at which organisms experience and interact with the environment. Here, we used geometric descriptors of phenotypic variation to derive a network representation of a complex morphological structure, the mammalian mandible, in terms of nodes and links. Then, by integrating the network representation and description with random matrix theory, we uncovered a modular organization for the mammalian mandible, which deviates significantly from an equivalent random network. The modules revealed by the network analysis correspond to the four morphogenetic units recognized for the mammalian mandible on a developmental basis. Furthermore, these modules are known to be affected only by particular genes and are also functionally differentiated. This study shows that the powerful formalism of network theory can be applied to the discovery of modules in complex phenotypes and opens the possibility of an integrated approach to the study of modularity at all levels of organizational complexity.
The model of Hastings and Powell describes a tritrophic food chain that exhibits chaotic dynamics... more The model of Hastings and Powell describes a tritrophic food chain that exhibits chaotic dynamics. The model assumes that the populations are homogeneously mixed, so that the probability that any two individuals interact is uniform and space can be ignored. In this paper we propose a spatial version of the Hastings and Powell model in which predators seek their preys only in a finite neighborhood of their home location, breaking the mixing hypothesis. Treating both space and time as discrete variables we derive a set of coupled equations that describe the evolution of the populations at each site of the spatial domain. We show that the introduction of local predator-prey interactions result in qualitatively distinct dynamics of predator and prey populations. The evolution equations for the predators involve averages over the local density of preys, whereas the equations for the preys involve double averages, where the local density of both preys and predators appear. Our numerical simulations show that local predation also leads to spontaneous pattern formation and to qualitative changes in the global dynamics of the system. In particular, depending on the size of the predation neighborhoods, the chaotic strange attractor present in the original model of Hastings and Powell can be replaced by a stable fixed point or by an attractor of simpler topology. (M.A.M. de Aguiar).
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Papers by Marcus Aguiar