In the first phase, the distribution moves gradually to higher g in the genotype space, keeping a unimodal shape, which indicates the gradual evolution of the quasi-species. The branching starts after the movement stops. The final states of the three figures in Fig. However, features of evolutionary branching dynamics are different. The moving distances of genotype distributions before the branchings begin vary with the intensity of fluctuation, which is characterized by the positions of distributions in the genotype space at the end of the first phase.
In Fig. At the no-branching conditions, only the first phase appears, and the two lines coincide.
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This evolutionary branching is induced not only by the periodic fluctuation but also stochastic fluctuations with Poisson process. It indicates the generality of the branchings induced by fluctuation. Hereafter, we mainly discuss about the system with the periodic fluctuation, because of its simplicity. Data from the evolutionary simulations are plotted with lines. The representative genotypes at the end of the first phase are plotted with cyan dashed lines, and the representative genotypes at the final state are plotted with blue continuous lines.
Characteristic values of PIPs discussed below , g c and g ex , are plotted with red squares and green circles, respectively.
Evolutionary branching induced by a stochastic fluctuation. Instead of using eq.
This simulation indicates that the stochastic fluctuation also induces the clear branching. Based on the given trade-off relation in parameters of the Monod equation, the finally resident quasi-species are characterized as the specialist adapted for the rich nutrient condition quasi-species with larger g , and as the generalist adapted for wider range of the nutrient condition quasi-species with smaller g.
Interestingly, the lengths of growing periods and the lengths of decreasing periods are almost the same between the both quasi-species as observed in Fig. Therefore the time-sharing of resource niche  does not seem to happen at the coexisting state.
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Instead, the growth rates and decrease rates in these phases are different from each other: quasi-species with larger g has higher growth and decrease rates, and quasi-species with smaller g has lower rates. The evolutionary branchings reported here is driven by the competition in population dynamics.
It is a frequency-dependent selection process  , and the evolutionary invasion analysis proposed by Geritz et al. Here, we introduce the technique and analyze our system to describe the processes of evolutionary branchings. We start the analysis with the non-fluctuating condition, which corresponds to the situations before the periodic fluctuations are added in the three simulations in Fig. Then, we analyze the fluctuating condition by using the pairwise-invasibility plot PIP , which is a tool in the analyzing method.
This raises the total consumption of the resource, and the resource concentration decreases. The monotonic increase of the Monod equation guarantees these processes.
The resident genotypes are replaced with genotypes with lower equilibriums sequentially until genotype g c is reached, and the resource concentration comes to. When fluctuation in the resource supply is introduced, the above simple discussion is not directly applicable. This is because that the growth or decay of populations depends on oscillation patterns of the resource concentration as an example of the oscillation profile, see Fig.
Such a situation is in contrast to the above condition, where only the static value of s determines the growth rate. In other words, the variety of oscillation profiles at the fluctuated conditions keeps off from having the simple order relation among genotypes and leads the increase of species diversity.
Unit 4: Ecosystems // Section 8: Evolution and Natural Selection in Ecosystems
To analyze the fluctuating conditions we introduce the pairwise-invasibility plot PIP , which gives the relation between genotypes and enables us to describe the evolutionary dynamics with the process of invasion and annihilation. PIP is constructed with the following procedure.
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The PIPs corresponding to Fig. Vertical lines on g c red solid lines pass through the shaded regions, which denotes genotypes invasible to g c. Here we investigate the structures of the PIPs and use them to illustrate evolutionary dynamics. The PIPs are constructed with two boundary lines. The other is a curved line 7 the shape of which determines the characteristics of PIP. First, we refer to the genotype at the intersection between lines as g c marked with red squares in Fig.
Any genotype lower than g c is invaded by genotypes higher than it area above the diagonal line is shaded and any genotype higher than g c is invaded by genotypes lower than it area below the diagonal line is shaded. This indicates that, as part of the evolutionary process, a resident genotype is invaded and replaced by genotypes closer to g c one after another.
It converges to g c. Therefore g c is called the convergent stable genotype . The convergent process corresponds to the gradual evolution of the quasi-species at the first phase in simulations. It stops when the representative genotype agrees with g c. The agreement between them is shown in Fig. This means that there are genotypes invasible for the population of g c , and this invasibility promotes evolutionary branching.
At the end of the first phase the population distributes around g c in the genotype space. If the tail of the distribution covers the region of invasible genotypes, the population of these genotypes grow and form another branch. Therefore, the evolutionary branching succeeds as the second phase. The features of branching dynamics are characterized by the relation between g c and genotypes invasible to g c.
Correspondingly in Fig. Thus, the presence and position of the extreme values inform us about the bifurcation properties of the system.
At the onset of the branching range, one quasi-species coincides with g c and the other coincides with g ex. If the gradient is positive corresponding to Fig. Therefore g c is locally unstable and the branching proceeds independent form the magnitude of the diffusion in genetic space D and the threshold of the population discreteness. In these cases, genotypes distant from g c are invasible, and whether they invade or not depends on the values of D and the threshold of the population discreteness.follow url
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If D is large enough and the threshold is small enough, the tail of the genotype distribution centered at g c brings the invasible genotypes, which invade and make another branch. Since the sign of the gradient changes at g ex s, the relation between g c and g ex s tells us which types of branching occurs. If g c has the value intermediate between both g ex s corresponding to Fig. On the other hand, if the g c is larger or smaller than both g ex s corresponding to Fig.
The figure 4 indicates the ranges of fluctuation intensity where the g c is locally unstable and the evolutionary branching occurs independent from D and the threshold of the population discreteness: 0. In summary, evolutionary branching dynamics induced by environmental fluctuation is reported in a model of ecosystems competing for a single resource. The evolutionary invasion analysis is applied to illustrate the evolutionary dynamics. Previous studies  ,  have reported the coexistence of two species with given parameters in the presence of fluctuation.
However, the occurrence of the evolutionary branching as the response to the environmental fluctuation and the evolutionary stability of the coexisting state have remained open questions. Here, we give clear demonstrations of them. The results also supports the general discussions which propose that the temporal fluctuation of environment increase the species diversity of the ecological systems  , . Moreover, we checked that some stochastic fluctuations with Poisson process in resource supply also induce the evolutionary branching and coexistence. It indicates the generality of the branching reported here.
Here we use a model of a microbial ecosystem to show the evolutionary branching induced by the environmental fluctuation. However, these phenomena are not specific to the model of microbial systems. In the following, we discuss the general mechanisms how the environmental fluctuation increases the species diversity. MacArthur and Levins  indicated that the number of independently adjustable environmental parameters corresponds to the maximum number of coexisting species.
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