Temporal inactivation enhances robustness in an evolving system

We study the robustness of an evolving system that is driven by successive inclusions of new elements or constituents with m random interactions to older ones. Each constitutive element in the model stays either active or is temporarily inactivated depending upon the influence of the other active elements. If the time spent by an element in the inactivated state reaches TW, it gets extinct. The phase diagram of this dynamic model as a function of m and TW is investigated by numerical and analytical methods and as a result both growing (robust) as well as non-growing (volatile) phases are identified. It is also found that larger time limit TW enhances the system’s robustness against the inclusion of new elements, mainly due to the system’s increased ability to reject ‘falling-together’ type attacks. Our results suggest that the ability of an element to survive in an unfavourable situation for a while, either as a minority or in a dormant state, could improve the robustness of the entire system.


Introduction
The robustness of a system with many interacting elements or constituents under successive addition of new elements is an essential question for understanding the behaviour of various & 2019 The Authors. Published by the Royal Society under the terms of the Creative different types of directed influences between the pairs of species. The strength of the influence of species j on species i is denoted by the weight of the unidirectional link from node j to node i, i.e. a ij . These weights can be either positive or negative. Each species has its 'fitness', which is simply given by the sum of its incoming interactions from other species in the system, i.e. f i = P incoming j a ij . A species can survive as long as its fitness is greater than zero. The species with non-positive fitness, which in our previous models [7,9] went instantaneously extinct, will in the present model be inactivated after its fitness-dependent waiting time t ¼ e f , i.e. species in worse situation is inactivated faster. The inactivated species loses its influence on other species; thus, we will neglect the links out of those for the calculation of fitness. If the surrounding community of an inactivated species changes and the fitness of an inactivated species becomes positive, the species is reactivated (waking up from dormancy). The waiting time of this reactivation process is also assumed to be fitness-dependent: t ¼ e 2f . The slowest process among the microscopic dynamics is the inactivation and reactivation of solitary species (f ¼ 0). The duration of these processes, t ¼ 1, gives the unit of time to this otherwise time-scale-less model. Although it is known that some species can maintain its dormancy for quite a long time [27], the period has generally a limit. In the following, we introduce a uniform time-limit parameter T W . A species that has spent T W of continuous time in the inactive state with non-positive fitness gets extinct. The extinct species and its incoming and outgoing links are removed permanently. Note that the present model with dormancy reduces to the original model [7] at T W ¼ 0. A pseudo-code style description of the entire dynamics is available in appendix A.
An example of temporal evolution of the system is shown in figure 2. If all the species are in active state and have positive fitnesses, nothing will happen. Therefore, we call such a state a persistent state. In the previous models [7,9], we added a new species every time the community has reached a persistent state. This corresponds to a low-introduction (mutation, invasion, etc.) rate limit. In the present model, however, it is also possible that the system relaxes to a limit cycle and never reaches a stationary persistent state (figure 3). Therefore, we need a new parameter for the time interval of the species introduction, T int . In the following, we take a long interval: T int ¼ 100 to keep a low-introduction rate, unless otherwise noted.
Every time after finding a persistent state or elapsed time T int , we proceed to the next time step by adding a new species with m interactions into the system. The m interacting species are chosen at random from the resident species with equal probability and the directions (incoming or outgoing) are also determined at random. The link weights are again assigned at random from the standard normal distribution.

Results
Following the approach of our previous study [7], we assess the robustness of the emergent system by the long-term trend of the system size, i.e. the number of species, under the successive introduction of new Figure 1. Introduction of the inactive state (dormancy) before the extinction, to our graph-dynamics framework. Less fit species is inactivated faster, and better fit species in inactive state is reactivated faster. The time limit of dormancy till extinction is, in contrast, uniformly set to T W . royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181471 species which, in terms of the directions and the weights of its interactions, has neutral effect on growth. In our original model without any dormant mechanism [7], the system can grow limitlessly; thus, it is robust enough against the inclusion of new species if the number of interactions given for each newly introduced species, m, is kept within a moderate range, i.e. 5 m 18. By contrast, the system with m outside this range, keeps fluctuating with a finite size. These fluctuations may lead to the extinction of the entire system, and the lower the mean level is, the higher is the probability for such an event. To avoid this possibility, we adopt an incubation rule when the system size becomes smaller than the initial system size N 0 . Under the incubation rule, we let totally isolated species (i.e. (a) Introduction of a new species (red), which makes the fitness of two species (orange and magenta) negative. Each of these two species will be inactivated after its fitness-dependent duration: t = e f i . (b) Inactivation of the species with worse fitness (orange) takes place first and then the other species (magenta) is inactivated, which makes the fitness of another species (green) nonpositive. Inactivated species is given T W of waiting time till it will go extinct. (c) Green species is inactivated before any of other inactive species goes extinct. This change makes the fitness of the inactive species (magenta) positive. (d ) Magenta species is reactivated after a fitness-dependent waiting time t = e f i . Meanwhile, the orange species have spent T W of time in the inactivated state and hence gone extinct: the orange species and the interactions from and to it are deleted. (e) Green species goes extinct. This does not change the sign of fitness of any species in the community. Therefore, after the extinction of green species, the system finally reaches a new persistent state, i.e. all the species are in the active state and have positive fitnesses. Nothing will happen for a community in a persistent state, until the next new species is introduced at t þ T int .
royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181471 the active state or inactive state. This treatment prevents the total collapse of the system and provides the system with many more opportunities to search for growth from different initial conditions. For sufficiently large initial system size, typically N 0 ! 100, the limitless growth and finite size fluctuation behaviour are confirmed to be independent of the initial network structure. Therefore, we call the former behaviour taking place in the 'diverging phase' and the latter in the 'finite phase' of the parameter space. The temporal evolution of the system size of the present model with m ¼ 25 is shown in figure 4. Inheriting the nature of the original model [7], the system with short dormancy limit T W is found to be in the finite phase. However, as T W increases (to the value T W ¼ 0.3) the typical system size shows a clear increase yet it stays finite, and for T W ¼ 0.4 and above the system has crossed a certain threshold to show diverging behaviour. This clearly illustrates that our newly introduced parameter T W , the time limit for the continuous dormancy, can change the robustness of the system. Next we will explore the whole phase diagram with systematic computer simulations by scanning through the m versus T W parameter space. The obtained phase diagram is shown in figure 5, where it is seen that the introduction of dormancy and revival processes broaden the diverging phase. While    this effect turns out to be larger for longer dormancy time limit T W , yet it is not possible to get the system with very dense interactions (m ! 28) to the diverging phase. It should also be noted that, as shown in appendix A, the basic network characteristics of the emergent systems are not so much dependent on T W and not deviated from that of Erdó´s-Rényi random graph, indicating the almost random network structure as in the previous models [7,9].
The main mechanism of this enforcement is the rejection of 'falling-together-attacks'. To illustrate this, let us consider a situation that a negative link weight (2a) is added to a resident species by a newly introduced species, which has zero or negative fitness value, 2b ( figure 6).
In the original model [7] and in the present model with T W ¼ 0, in which the least fit species goes extinct first, the attacked resident species and the new species sequentially go extinct for f 2 a , 2b and otherwise only the new species goes extinct (i.e. is rejected). Especially for the newly introduced species with no incoming links (b ¼ 0, solitary attack), every attack strong enough (f , a) can kill the resident species before the newly introduced attacker species goes extinct.
In the present model with T W . 0, the situation is different as the resident species has another chance to reject such a falling-together attack. The rejection happens if the resident species can survive in the inactivated state until the newly added species stays inactivated. The condition for this type of dynamics is as follows: f À a , Àb , ln (e fÀa + T W ): (3:1) Therefore, even a strong attack (f . a) by a solitary new species (b ¼ 0) is rejected if T W . 1 2 e f2a . And if T W ! 1, i.e. the limit of the dormancy period is long enough, even the solitary attacks never become successful. Note that the rejection acts perfectly in a special case of m ¼ 1, because in this situation every inclusion of new species corresponds to either a solitary attack or an attachment of species with no outgoing link. Therefore, even for this most sparse condition, large T W drives the system with a mutually supporting community core to grow infinitely in size. However, such a growth is highly dependent on the initial condition (if there is no core in the initial network, the system collapses) which is out of the scope of this study. Thus, we excluded this case from the phase diagram. The increment of probability to reject falling-together-attacks directly contributes to the growth rate of the system, v ¼ N(t)/t. A rough estimate of it near the upper phase boundary (m 18) predicts a linear increase of the rejections to T W for the small T W regime (see appendix A for details), which is confirmed in the simulation ( figure 7). The observed contribution of the additional rejections to the system's growth rate, Dv T W /8, predicts the slope of the phase boundary to behave as Dm* 20 T W . This is found to be consistent with the phase portrait.
The effect of rejections in the sparse regime (m 4) needs to be estimated differently. This is because the probability to have a solitary attack is larger. What is more significant, however, is the fact that the resident community has a sparse network structure, which in turn is very prone to a loss of certain species and can cause a cascade of extinctions of species supported by that species. Therefore, the effect of the increased chance of rejection can be more drastic. It is also possible that the structure of the emergent networks is changed, although the basic network characteristics (see appendix A) and the well-kept distributions of extinction cascade size suggests it to be negligible at least for m ¼ 4

Summary and discussion
We have studied the robustness of an evolving system against successive inclusions of new elements or constituents, each with an ability to survive temporarily under unfavourable conditions in the state of being inactive. It is found that the introduction of the inactivation and revival processes broadens the phase the systems stays robust. This reinforcement of the emerging system is mainly due to its increased ability to reject falling-together type attacks. It should be noted that the broadening of the robust phase has a limit: systems with m ! 28 stay in the finite phase even at T W ¼ 1, where the rejection probability reaches its maximum. The short-term rejection process, in which a possible extinction of a species caused by the attack from a species with poor fitness is altered by the extinction of the attacker, can be regarded as a simplified dynamics in a class of population dynamics models [13][14][15]22]. Because another type of interaction form, namely the ratio-dependent interaction [28], is known to reduce to our previous model [29], the extension of the model in this study has broadened the applicability of our theoretical framework. Similar to our earlier results [7,9], we have found that the number of interactions per species limits the system's robustness. There are empirical findings in support to this observation [30].
As for the modelling in general the population dynamics models based on differential or difference state equations are able to describe rich evolutionary patterns following periodic and even chaotic trajectories, as observed in nature [31,32]. However, this approach is generally computationally so costly that larger system sizes and longer time scales could not be studied. In order to circumvent these problems, we have taken a network-based approach, which is able to describe the dynamics of the system over much longer evolutionary time scale. Although our present analysis covers up to the long-dormancy time limit (T W ¼ 1) in terms of the resulting short-term rejection process, far longer dormancy limit (T W ) T int ) could bring new phenomena. Under such condition, inactive species can survive evolutionary time scale during which new species are introduced and that change the community. In some cases and for various kinds of systems, such as biological, social and economic systems, it may be important to consider such long dormancy periods [33]. Also, the effect of bidirectionality [9] of the interaction should be examined, because it is expected to make the emergent system show limit cycles more frequently. These two regimes, although that requires heavier computation power, will reveal new phenomena and will better bridge with the continuous time dynamics models. Extending our approach so that some aspects of short-term dynamics of more complex models are kept, with further spacial extension focusing on some aspects hardly accessible by traditional methods, is a promising way to treat evolutionary problems better [34,35].
royalsocietypublishing.org/journal/rsos R. Soc. open sci. 6: 181471 (4) Find the shortest time to the next event in the system: dt Ã j = min {dt i }. (5) Time translation of the system from t to t + dt Ã j (i) Update the system time t = t + dt Ã j (ii) Update the time counters: (v) Update the time for the next species introduction: T next ¼ T next þ T int . (7) Recalculate the fitness: go back to step (1).

A.2. Estimation of the rate of the additional rejections and its effect
Here we first roughly estimate the increment of the chance to reject such falling-together-attack which directly contributes to the growth rate of the system, v ¼ N(t)/t, near the upper phase boundary (m 18). In the vicinity of the phase boundary in the dense regime, an inclusion of new species causes one strong attack (f , a) event in average. The distribution of f 2 a is given by the negative side of the convolution: where f(x) and G(s, x) represent the equilibrium fitness distribution of the emergent system and the Gaussian distribution with its standard deviation s, respectively. The distribution of the fitness of newly added species, 2b, is well approximated by the negative half side of the Gaussian distribution is also consistent with the very rough estimation above.
Taking the linear slope of the system's intrinsic growth rate to m obtained from the observed growth rates