# Universal Evolutionary Model for Periodical Species

Authors: Eric Goles; Ivan Slapnicar; Marco A. Lardies
Journal: Complexity
 (2021)

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## Abstract

Real-world examples of periodical species range from cicadas, whose life cycles are large prime numbers, like 13 or 17, to bamboos, whose periods are large multiples of small primes, like 40 or even 120. The periodicity is caused by interaction of species, be it a predator-prey relationship, symbiosis, commensalism, or competition exclusion principle. We propose a simple mathematical model, which explains and models all those principles, including listed extremal cases. This rather universal,qualitative model is based on the concept of a local fitness function, where a randomly chosen new period is selected if the value of the global fitness function of the species increases. Arithmetically speaking, the different interactions are related to only four principles: given a couple of integer periods either (1) their greatest common divisor is one, (2) one of the periods is prime, (3) both periods are equal, or (4) one period is an integer multiple of the other.


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## Body

## 1. Introduction

Life-history diversity is a remarkable feature of living species and underlies fundamental evolutionary questions [1]. Periodical species are well known for their mass and synchronous reproduction [2]. The use of an appropriate model to evaluate synchronicity in periodical species depends first on the life-history strategies of the species. For example, some species have a life-history with discrete nonoverlapping generations; that is, there are no adult survivors from one generation to the next. Such species includes annual plants, annual insects, salmon, periodical cicadas, and bamboos species [3]. Once the adults reproduce, they perish, and the future of the population is based on the dormant or juvenile stage of the organism. In periodical species, the population emerges synchronously from the ground or benthos (cicadas, bamboos, red or brown tides of phytoplankton species) as juveniles or adults. The growth rates before emerging in periodical cycle species are obviously affected by the environment being the main driver for maintaining a minimum viable population (for example, extinction probability), the interspecific interactions with predators (including parasitoids), symbiosis, interspecific competition, and/or disturbances in the habitat [4–7]. Furthermore, the nature of these interactions can vary depending on the evolutionary context and environmental conditions in which they occur [8]; nevertheless, the synchrony is maintained despite abrupt changes in the environment [9, 10].Dilucidating, in which natural forces drive these extraordinary periodical cycles, is a central challenge in ecology. To understand these evolutionary processes, a broader view is needed of the properties of multiscale spatiotemporal patterns in species–environment interactions [11]. Periodical species are plants or animals that emerge in nature every T>1 years, like some few species of cicadas (every 3, 13, and 17 years) or bamboos (2, 3, 5, 15, 32, 60, and 120 years), and other flowering plants [12]. Density-dependent selection is the simplest form of feedback between an ecological effect of an organism’s own making (crowding due to sustained population growth) and the selective response to the resulting conditions [13]. Density-dependent fitness and density-dependent selection are critical concepts underlying ideas about adaptation to biotic selection pressures and the coadaptation of interacting species. To understand those periods (and other related processes), several models have been proposed [14]. In most of them, one of the relevant assumptions is the “satiation hypothesis”; that is, organisms have to emerge in very large numbers in order to satiate predators (birds, rats, etc.), which is consequently linked to climate conditions [12, 15] but also to reduction in ecosystem productivity [10]. Those assumptions explain synchronicity but not necessarily T>1 life cycles. Since the usual life cycle of those organisms is annual, the question is how it may evolve to a T>1 period? Several evolutionary pressures have been proposed to explain periodical life cycles. Some years ago, a deterministic discrete populational model was presented as a mechanism that is related with systematic low temperatures in the ice ages in Pleistocene [16, 17]. In this case, under the cold temperature pressure, selected cicadas are those with large periods underground during the metamorphosis stage in order to avoid low temperatures (but see also [18]). Other theories regarding stabilizing selection provide explanations for the existence of synchronous seeding in bamboos [19]. From this latter assumption, they proposed a model such that, from the usual synchronously annual emergence of seeds, a mutation with period T>1 may appear. However, some problems remained: Why periodical cicadas have prime life cycles? Is discrete multiplication the explanation that some flowering intervals evolved in bamboos have small prime cycles while some have very large nonprimes life cycles? Some authors have proposed a very simple model for cicadas prime life cycles by assuming the existence of a periodical predator [20]. In this context, it was proved that the fixed points for the nonextinct periodical cicadas are necessarily prime numbers. Nevertheless, simulations were done, which exhibit convergence to prime numbers only by accepting mutations in a narrow temporal scale (for instance ±one year concerning the current life cycle). Other models take into account an hybridization hypothesis, which entails that life cycle evolution has a phylogenetic component, where evolution of periodicity in species was due to glacial cooling during the Pleistocene and the selection for prime-numbered cycles via hybrid breakdown of nonprime-numbered cycles [17]. That is, if the life cycle of two species of prey is prime numbers, T and T′, so that they encounter only every multiple of T×T′ years, this diminishes drastically the probability of hybridization, which will produce other life cycles with less offspring [16]. However, it is important to remark that, in order to satisfy the hybridization hypothesis, it is enough to consider only relative primes life cycles [14]. Based on the several gaps of the models presented previously, it is required to develop a mathematical theory that includes the possible mechanism of how periodical life cycles are shaped by diverse kinds of interspecific interactions in an evolutionary context in nature.In this work, we propose a universal qualitative framework for species with synchronic periodical cycles, which is based on local fitness functions and the evolution of species interaction evaluating the extinction probability of species. Additionally, we describe the convergence of the proposed model and completely characterize its fixed points, leading to the conclusion that different interactions are related to only four principles. The paper is organized as follows: in section General Model, we define general model based on the notion of a fitness function. In section Classes of Fitness Functions, we classify all fitness functions that appear in our model and reduce them to 18 cases (fitness tuples), which need to be studied. In section Dynamics of the Model, we describe the dynamics of the model and state the evolution algorithm. In section Characterization of Fixed Points, we characterize all fixed points of the studied fitness functions in the case of bounded mutations. In section Numerical Simulations, we present numerical simulations for two cases of bounded mutations as well as more general “quantitative” fitness functions. In section Discussion, we discuss our results and some open questions, including small modifications with fixed points, which are very large periods. In section Conclusion, we draw our conclusions.


## 2. Model

### 2.1. General Model

Consider two species,C1 and C2, with an emergence period c1,c2∈ℕ, c1,c2≥2 (in years). Consider the interval T=c1⋅c2, and the emergence functions(1)χi:1,T⟶0,1,such that χit=1 if and only if the species Ci emerges in year t. Clearly, species C1 emerges exactly c2 times in the interval 1,T and, respectively, C2 emerges exactly c1 times.Let us now define the (local) fitness functions associated toC1 and C2 as(2)fi:0,1×0,1⟶−1,0,1,i=1,2,such that f10,∗=f2∗,0=0; that is, if species is dormant, the value of its fitness function is zero. If the species is emergent, the value of its fitness function with respect to the other species may be 1, 0 or −1 (good, neutral or bad). In biological terms, this fitness function is an expected fitness, the probability of success given each periodical species’ emergence. Furthermore, expected fitness is directly related to per capita population growth rate; that is, if fitness function is 1, population increases in size; if function is 0, population maintains its size; and if the fitness function is −1, population decreases in size to extinction. This simple model is qualitative, but it captures all interesting types of behavior.For given periodsc1 and c2, the global (cumulative) fitness function over the interval 1,T is obtained by simply adding yearly values of local fitness function and dividing them by the number of years, in which the species appears in this interval:(3)F1c1,c2=1c2∑t=1Tf1χ1t,χ2t,F2c1,c2=1c1∑t=1Tf2χ1t,χ2t.


### 2.2. Classes of Fitness Functions

Given two species,C1 and C2, and their local fitness functions, f1 and f2, respectively, the only interesting situations are when the species are emergent. Therefore, the only cases, which need to be considered, are given by the 4-tuple(4)v≡ν1,ν2,ν3,ν4=f11,0,f20,1,f11,1,f21,1∈−1,0,14.We shall call such 4-tuplefitness tuple or just tuple. Consequently, we have a total of 34=81 possibly different tuples v. We reduce this number in three steps to only 18 tuples that need to be studied as follows. First, we eliminate equivalent tuples (two tuples are equivalent when they are obtained by simply exchanging species C1 and C2, that is ν1,ν2,ν3,ν4≡ν2,ν1,ν4,ν3). Then, we are left with v to 45 tuples, which we divide into four classes (for the class D we also take into account the symmetry when one exchanges species, that is, a,b,c,d=b,a,d,c):(5)A=a,a,b,b|a,b∈−1,0,1=−1,−1,−1,−1,−1,−1,0,0,−1,−1,1,1,0,0,−1,−1,0,0,0,0,0,0,1,1,1,1,−1,−1,1,1,0,0,1,1,1,1,B=a,a,−1,0,a,a,−1,1,a,a,0,1|a∈−1,0,1=−1,−1,−1,0,−1,−1,−1,1,−1,−1,0,1,0,0,−1,0,0,0,−1,1,0,0,0,1,1,1,−1,0,1,1,−1,1,1,1,0,1,C=−1,0,a,a,−1,1,a,a,0,1,a,a|a∈−1,0,1=−1,0,−1,−1,−1,1,−1,−1,0,1,−1,−1,−1,0,0,0,−1,1,0,0,0,1,0,0,−1,0,1,1,−1,1,1,1,0,1,1,1,D=a,b,c,d|a≠b∧c≠d−1,0,−1,0,−1,0,0,−1,−1,0,−1,1,−1,0,1,−1,−1,0,0,1,−1,0,1,0,−1,1,−1,0,−1,1,0,−1,−1,1,−1,1,−1,1,1,−1,−1,1,0,1,−1,1,1,0,0,1,−1,0,0,1,0,−1,0,1,−1,1,0,1,1,−1,0,1,0,1,0,1,1,0.In the second step, we eliminate tuples without biological interest or meaning:(6)−1,−1,−1,−1,−1,−1,0,0,0,0,−1,−1,0,0,0,0,1,1,0,0,1,1,1,1,−1,−1,−1,0,−1,−1,−1,1,−1,−1,0,1,0,0,−1,0,0,0,0,1,−1,0,−1,−1,−1,1,−1,−1,−1,0,0,0,−1,1,0,0,0,1,0,0,0,1,1,1,−1,0,−1,0,−1,0,−1,1,−1,1,−1,0,−1,1,−1,1,0,1,0,1.For example, for the tuple−1,0,−1,0, the species does not have any positive evolution pressure, and the other species is completely indifferent.In the third step, we eliminate tuples, for which one of the fitness functions is always negative according to the following lemma:Lemma 1.
For the tuples(7)0,1,−1,−1,0,0,−1,1,0,1,−1,1,0,1,−1,0,0,−1,−1,0,0,1,−1,0,−1,0,1,−1,one of the global fitness functions is always negative.
See Appendix.
Therefore, the final set of tuples to study,V, consists of the 18 tuples listed in Table 1. We call such tuples relevant fitness tuples and we number them for further reference.
Each tuple may be represented by its respective ecological graph as in Figure1. By inspecting corresponding ecological graphs, we classify tuples from Table 1 in four classes. The first class, V1, is the set of all tuples, for which the corresponding graph is not strongly connected:(8)V1=v∈V|f11,1=0∨f21,1=0=v7,v10,v11,v13,v14,v15,v16,v17,v18.
The other three classes are(9)V2=v∈V|f11,1=f21,1=1=v1,v2,v3,v4,v5,V3=v∈V|f11,1=f21,1=−1=v6,V4=v∈V|f11,1⋅f21,1=−1=v8,v9,v12.Table 1
Relevant fitness tuples.
v1=−1,0,1,1v2=−1,1,1,1v3=0,1,1,1v4=0,0,1,1v5=−1,−1,1,1v6=1,1,−1,−1v7=1,1,−1,0v8=1,1,1,−1v9=−1,1,1,−1v10=−1,1,1,0v11=1,0,−1,0v12=0,1,1,−1v13=0,1,1,0v14=1,1,1,0v15=0,−1,1,0v16=−1,0,1,0v17=1,−1,−1,0v18=1,−1,1,0Figure 1
General ecological graph, where two emergent species,C1 and C2, interact with local fitness functions and both species coexist.


### 2.3. Dynamics of the Model

We describe the evolution of speciesC1 and C2. Given initial periods c1,c2∈ℕ,c1,c2≥2, the first iteration of the evolution game is as follows: for the species C1 we propose random local change of its period c1 to new period c1′∈ℕ,c1′≥2. If the value of the global fitness function of the species C1 has improved, that is, if F1c1′,c2>F1c1,c2, the period c1′ is accepted as the new period, c1←c1′, and the value of the global fitness function of the other species, C2, is recomputed. The analogous procedure is now applied to the species C2. Such iterations are repeated until there is no change. The pseudocode of the algorithm is given in Algorithm 1.Algorithm 1: Evolution dynamics.
procedure evolution (c1,c2,v) ⊳ c1,c2∈ℕ,c1,c2≥2T=c1⋅c2ComputeF1c1,c2 and F2c1,c2loopchoosec1′ at random ⊳ c1′≥2T′=c1′⋅c2computeF1′c1′,c2=1/c2∑t=1T′f1χ1t,χ2tifF1′c1′,c2>F1c1,c2then ⊳ the fitness of C1 improvedc1←c1′ ⊳ accept the new periodrecomputedF2c1,c2end ifchoosec2′ at random ⊳ c2′≥2T′=c1⋅c2′computeF2′c1,c2′=1/c1∑t=1T′f2χ1t,χ2tifF2′c1,c2′>F2c1,c2then ⊳ the fitness of C2 improvedc2←c2′ ⊳ accept the new periodrecomputedF1c1,c2end ifend loopend procedureWe will also assume that the evolutionary game changes each species in a mutation interval−p,p=−p,−p+1,…,−1,0,1,…,p; that is, the arbitrarily far moves do not exist in the algorithm. More precisely, the new periods c1′ and c2′ are chosen as follows:(10)c1′=c1+k,c2′=c2+l,k,l∈−p,p.We say that a couple of periodsc1,c2 for the given tuple v are a p-fixed point if and only if ∀k,l∈−p,p(11)F1c1+k,c2≤F1c1,c2,F2c1,c2+l≤F2c1,c2.In this context, fixed points are related to the interval of possible time jumps. We also speak ofglobal fixed point when the above is true for any p∈ℕ.


### 2.4. Characterization of Fixed Points

For the next results, we will consider the evolution algorithm forp=1, that is, in −1,1 mutation space. In particular, by “local fixed point” we mean 1-fixed point. The results for p-fixed points are similar (see Numerical Simulations and Discussion).The first results are related with the relations between the sets of fixed points in any of the classesVi, i=1,2,3,4. For that, let us define the set of fixed points as follows:(12)Ak=c1,c2|c1,c2∈ℕ,and it is an fixed point for the tuplevk.The relations between the sets are as follows.Lemma 2.
For the tuples in the classV1 we have(13)A7⊊A11,A10=A13⊊A14,A15⊊A16,A10⊊A16,A17⊊A11,A17⊊A18.
For the tuples in the classV2 we have(14)A1=A4=A5⊊A2=A3.
For the tuples in the classV4 we have(15)A9=A12⊊A8.
See Appendix.
Using Lemma2, we eliminate tuples v3, v4, v5, v12 and v13 from Table 1. We shall call the remaining tuples inside each class representative tuples. Now, we characterize the fixed points inside each class.Proposition 1.
For the representative tuples in the classV1 the structure of fixed points is as follows:(i)
Forv7 there exist global fixed points such that gcdc1,c2=1, and there exist infinitely many local ones.(ii)
Forv10 there exist infinitely many local fixed points, but no global ones.(iii)
Forv11 there exist global fixed points such that gcdc1,c2=1, as well as infinitely many local ones.(iv)
Forv14 there exist global fixed points such that gcdc1,c2=1, as well as infinitely many local ones.(v)
Forv15, the couples c1,c2 such that c1=c2 are global fixed points. Further, there exist infinitely many local ones.(vi)
Forv16, the couples c1,c2 such that c2∣c1 are global fixed points, and there exist infinitely many local ones.(vii)
Forv17 there are no global fixed points, but there are infinitely many local ones. For instance, couples c1,c2 such that c1 is a prime number and c2<c1−1 are local fixed points.(viii)
Forv18, the couples c1,c2 such that c1∣c2 are global fixed points, and there exist infinitely many local ones.
See Appendix.Proposition 2.
For the representative tuples in the classV2 we have(i)
v1 admits the global fixed points c1,c2 such that c1=c2. Further, there exist also infinitely many local fixed points.(ii)
Forv2, any couple c1,c2 such that c2|c1 is a global fixed point. Further, there exist infinitely many local fixed points.
See Appendix.Proposition 3.
For the classV3=v6, every c1,c2 such that gcdc1,c2=1, is a global fixed point. There exist infinitely many local fixed points.
See Appendix.Proposition 4.
For the representative tuples in the classV4 we have(i)
v8 admits global fixed points c1,c2 such that gcdc1,c2=1. Further, it admits infinitely many local fixed points.(ii)
v9 only admits local fixed points: if c1,c2 is such that c2 is prime and c2∉c1−1,c1+1, then it is a local fixed point. Further, the species C1 disappears. Also, there exist other infinite families of local fixed points.
See Appendix.


## 2.1. General Model

Consider two species,C1 and C2, with an emergence period c1,c2∈ℕ, c1,c2≥2 (in years). Consider the interval T=c1⋅c2, and the emergence functions(1)χi:1,T⟶0,1,such that χit=1 if and only if the species Ci emerges in year t. Clearly, species C1 emerges exactly c2 times in the interval 1,T and, respectively, C2 emerges exactly c1 times.Let us now define the (local) fitness functions associated toC1 and C2 as(2)fi:0,1×0,1⟶−1,0,1,i=1,2,such that f10,∗=f2∗,0=0; that is, if species is dormant, the value of its fitness function is zero. If the species is emergent, the value of its fitness function with respect to the other species may be 1, 0 or −1 (good, neutral or bad). In biological terms, this fitness function is an expected fitness, the probability of success given each periodical species’ emergence. Furthermore, expected fitness is directly related to per capita population growth rate; that is, if fitness function is 1, population increases in size; if function is 0, population maintains its size; and if the fitness function is −1, population decreases in size to extinction. This simple model is qualitative, but it captures all interesting types of behavior.For given periodsc1 and c2, the global (cumulative) fitness function over the interval 1,T is obtained by simply adding yearly values of local fitness function and dividing them by the number of years, in which the species appears in this interval:(3)F1c1,c2=1c2∑t=1Tf1χ1t,χ2t,F2c1,c2=1c1∑t=1Tf2χ1t,χ2t.


## 2.2. Classes of Fitness Functions

Given two species,C1 and C2, and their local fitness functions, f1 and f2, respectively, the only interesting situations are when the species are emergent. Therefore, the only cases, which need to be considered, are given by the 4-tuple(4)v≡ν1,ν2,ν3,ν4=f11,0,f20,1,f11,1,f21,1∈−1,0,14.We shall call such 4-tuplefitness tuple or just tuple. Consequently, we have a total of 34=81 possibly different tuples v. We reduce this number in three steps to only 18 tuples that need to be studied as follows. First, we eliminate equivalent tuples (two tuples are equivalent when they are obtained by simply exchanging species C1 and C2, that is ν1,ν2,ν3,ν4≡ν2,ν1,ν4,ν3). Then, we are left with v to 45 tuples, which we divide into four classes (for the class D we also take into account the symmetry when one exchanges species, that is, a,b,c,d=b,a,d,c):(5)A=a,a,b,b|a,b∈−1,0,1=−1,−1,−1,−1,−1,−1,0,0,−1,−1,1,1,0,0,−1,−1,0,0,0,0,0,0,1,1,1,1,−1,−1,1,1,0,0,1,1,1,1,B=a,a,−1,0,a,a,−1,1,a,a,0,1|a∈−1,0,1=−1,−1,−1,0,−1,−1,−1,1,−1,−1,0,1,0,0,−1,0,0,0,−1,1,0,0,0,1,1,1,−1,0,1,1,−1,1,1,1,0,1,C=−1,0,a,a,−1,1,a,a,0,1,a,a|a∈−1,0,1=−1,0,−1,−1,−1,1,−1,−1,0,1,−1,−1,−1,0,0,0,−1,1,0,0,0,1,0,0,−1,0,1,1,−1,1,1,1,0,1,1,1,D=a,b,c,d|a≠b∧c≠d−1,0,−1,0,−1,0,0,−1,−1,0,−1,1,−1,0,1,−1,−1,0,0,1,−1,0,1,0,−1,1,−1,0,−1,1,0,−1,−1,1,−1,1,−1,1,1,−1,−1,1,0,1,−1,1,1,0,0,1,−1,0,0,1,0,−1,0,1,−1,1,0,1,1,−1,0,1,0,1,0,1,1,0.In the second step, we eliminate tuples without biological interest or meaning:(6)−1,−1,−1,−1,−1,−1,0,0,0,0,−1,−1,0,0,0,0,1,1,0,0,1,1,1,1,−1,−1,−1,0,−1,−1,−1,1,−1,−1,0,1,0,0,−1,0,0,0,0,1,−1,0,−1,−1,−1,1,−1,−1,−1,0,0,0,−1,1,0,0,0,1,0,0,0,1,1,1,−1,0,−1,0,−1,0,−1,1,−1,1,−1,0,−1,1,−1,1,0,1,0,1.For example, for the tuple−1,0,−1,0, the species does not have any positive evolution pressure, and the other species is completely indifferent.In the third step, we eliminate tuples, for which one of the fitness functions is always negative according to the following lemma:Lemma 1.
For the tuples(7)0,1,−1,−1,0,0,−1,1,0,1,−1,1,0,1,−1,0,0,−1,−1,0,0,1,−1,0,−1,0,1,−1,one of the global fitness functions is always negative.
See Appendix.
Therefore, the final set of tuples to study,V, consists of the 18 tuples listed in Table 1. We call such tuples relevant fitness tuples and we number them for further reference.
Each tuple may be represented by its respective ecological graph as in Figure1. By inspecting corresponding ecological graphs, we classify tuples from Table 1 in four classes. The first class, V1, is the set of all tuples, for which the corresponding graph is not strongly connected:(8)V1=v∈V|f11,1=0∨f21,1=0=v7,v10,v11,v13,v14,v15,v16,v17,v18.
The other three classes are(9)V2=v∈V|f11,1=f21,1=1=v1,v2,v3,v4,v5,V3=v∈V|f11,1=f21,1=−1=v6,V4=v∈V|f11,1⋅f21,1=−1=v8,v9,v12.Table 1
Relevant fitness tuples.
v1=−1,0,1,1v2=−1,1,1,1v3=0,1,1,1v4=0,0,1,1v5=−1,−1,1,1v6=1,1,−1,−1v7=1,1,−1,0v8=1,1,1,−1v9=−1,1,1,−1v10=−1,1,1,0v11=1,0,−1,0v12=0,1,1,−1v13=0,1,1,0v14=1,1,1,0v15=0,−1,1,0v16=−1,0,1,0v17=1,−1,−1,0v18=1,−1,1,0Figure 1
General ecological graph, where two emergent species,C1 and C2, interact with local fitness functions and both species coexist.


## 2.3. Dynamics of the Model

We describe the evolution of speciesC1 and C2. Given initial periods c1,c2∈ℕ,c1,c2≥2, the first iteration of the evolution game is as follows: for the species C1 we propose random local change of its period c1 to new period c1′∈ℕ,c1′≥2. If the value of the global fitness function of the species C1 has improved, that is, if F1c1′,c2>F1c1,c2, the period c1′ is accepted as the new period, c1←c1′, and the value of the global fitness function of the other species, C2, is recomputed. The analogous procedure is now applied to the species C2. Such iterations are repeated until there is no change. The pseudocode of the algorithm is given in Algorithm 1.Algorithm 1: Evolution dynamics.
procedure evolution (c1,c2,v) ⊳ c1,c2∈ℕ,c1,c2≥2T=c1⋅c2ComputeF1c1,c2 and F2c1,c2loopchoosec1′ at random ⊳ c1′≥2T′=c1′⋅c2computeF1′c1′,c2=1/c2∑t=1T′f1χ1t,χ2tifF1′c1′,c2>F1c1,c2then ⊳ the fitness of C1 improvedc1←c1′ ⊳ accept the new periodrecomputedF2c1,c2end ifchoosec2′ at random ⊳ c2′≥2T′=c1⋅c2′computeF2′c1,c2′=1/c1∑t=1T′f2χ1t,χ2tifF2′c1,c2′>F2c1,c2then ⊳ the fitness of C2 improvedc2←c2′ ⊳ accept the new periodrecomputedF1c1,c2end ifend loopend procedureWe will also assume that the evolutionary game changes each species in a mutation interval−p,p=−p,−p+1,…,−1,0,1,…,p; that is, the arbitrarily far moves do not exist in the algorithm. More precisely, the new periods c1′ and c2′ are chosen as follows:(10)c1′=c1+k,c2′=c2+l,k,l∈−p,p.We say that a couple of periodsc1,c2 for the given tuple v are a p-fixed point if and only if ∀k,l∈−p,p(11)F1c1+k,c2≤F1c1,c2,F2c1,c2+l≤F2c1,c2.In this context, fixed points are related to the interval of possible time jumps. We also speak ofglobal fixed point when the above is true for any p∈ℕ.


## 2.4. Characterization of Fixed Points

For the next results, we will consider the evolution algorithm forp=1, that is, in −1,1 mutation space. In particular, by “local fixed point” we mean 1-fixed point. The results for p-fixed points are similar (see Numerical Simulations and Discussion).The first results are related with the relations between the sets of fixed points in any of the classesVi, i=1,2,3,4. For that, let us define the set of fixed points as follows:(12)Ak=c1,c2|c1,c2∈ℕ,and it is an fixed point for the tuplevk.The relations between the sets are as follows.Lemma 2.
For the tuples in the classV1 we have(13)A7⊊A11,A10=A13⊊A14,A15⊊A16,A10⊊A16,A17⊊A11,A17⊊A18.
For the tuples in the classV2 we have(14)A1=A4=A5⊊A2=A3.
For the tuples in the classV4 we have(15)A9=A12⊊A8.
See Appendix.
Using Lemma2, we eliminate tuples v3, v4, v5, v12 and v13 from Table 1. We shall call the remaining tuples inside each class representative tuples. Now, we characterize the fixed points inside each class.Proposition 1.
For the representative tuples in the classV1 the structure of fixed points is as follows:(i)
Forv7 there exist global fixed points such that gcdc1,c2=1, and there exist infinitely many local ones.(ii)
Forv10 there exist infinitely many local fixed points, but no global ones.(iii)
Forv11 there exist global fixed points such that gcdc1,c2=1, as well as infinitely many local ones.(iv)
Forv14 there exist global fixed points such that gcdc1,c2=1, as well as infinitely many local ones.(v)
Forv15, the couples c1,c2 such that c1=c2 are global fixed points. Further, there exist infinitely many local ones.(vi)
Forv16, the couples c1,c2 such that c2∣c1 are global fixed points, and there exist infinitely many local ones.(vii)
Forv17 there are no global fixed points, but there are infinitely many local ones. For instance, couples c1,c2 such that c1 is a prime number and c2<c1−1 are local fixed points.(viii)
Forv18, the couples c1,c2 such that c1∣c2 are global fixed points, and there exist infinitely many local ones.
See Appendix.Proposition 2.
For the representative tuples in the classV2 we have(i)
v1 admits the global fixed points c1,c2 such that c1=c2. Further, there exist also infinitely many local fixed points.(ii)
Forv2, any couple c1,c2 such that c2|c1 is a global fixed point. Further, there exist infinitely many local fixed points.
See Appendix.Proposition 3.
For the classV3=v6, every c1,c2 such that gcdc1,c2=1, is a global fixed point. There exist infinitely many local fixed points.
See Appendix.Proposition 4.
For the representative tuples in the classV4 we have(i)
v8 admits global fixed points c1,c2 such that gcdc1,c2=1. Further, it admits infinitely many local fixed points.(ii)
v9 only admits local fixed points: if c1,c2 is such that c2 is prime and c2∉c1−1,c1+1, then it is a local fixed point. Further, the species C1 disappears. Also, there exist other infinite families of local fixed points.
See Appendix.


## 3. Numerical Simulations

We ran simulations to illustrate the behavior of the representative 4-tuples. A couple of periodsc1,c2 are a p-fixed point under simulation if, after thirty steps, neither of the species changes its period. We plot fixed points using the following convention: if c1,c2 is not a fixed point, the corresponding square on the plot is white; otherwise, it is red if both species survive, or gray if the species C1 disappears. The link to the complete code used to run simulations is given in Declarations.The plots in Figures2–7 depict the characteristic behavior of the qualitative model described in Propositions 1–4 (see also Discussion). For easier visual reference when interpreting the figures, Figure 8 shows canonical plots of three typical situations.Figure 2
Attractors for tuplesv7, v11 and v17 from Proposition 1. The tuples are of a “competition” type. Global fixed points for v7 and v11 are those with gcdc1,c2=1. In v17, the species C1 has a behavior like periodic cicadas with prime number fixed points. In this case, the species C2 disappears, since the fitness F2c1,c2 is always negative. Attractors are marked with red squares if both species survive and with gray squares if one of the species disappears.Figure 3
Attractors for tuplesv16 and v18 from Proposition 1. Here, one of the global fitness functions is constant (F2c1,c2=0 for v16 and F1c1,c2=1 for v18), while the other species has a bamboo-like behavior—it emerges as an integer multiple of the constant one. In both cases, there is a “commensalism” interaction.Figure 4
Attractors for tuplesv14 and v15 from Proposition 1. For v1, global fixed points are again those with gcdc1,c2=1, and for v15, global fixed points are those with c1=c2.Figure 5
Attractors for tuplesv1 and v2 from Proposition 2. The tuple v1 generates symbiosis of the two species; that is, they have to appear at the same time (here, global attractors are again those with c1=c2). The tuple v2 generates a “commensalism” relation: the period c1 has to be an integer multiple of c2, c2|c1, independently of c1 (a “bamboo” effect).Figure 6
Attractors for tuplev6 from Proposition 3. This is the case of two species, which cannot emerge together—they are mutually harmful. In this sense, the competition exclusion principle can originate local extinction in populations. Global fixed points are those with gcdc1,c2=1. Notice that the fixed points for fitness tuples v6, v7, v8, v11 and v14 are similar.Figure 7
Attractors for tuplesv8 and v9 from Proposition 4. Both tuples generate predator-prey interaction, with the species C2 being the prey. For v8, C1 does not depend on C2 (it may feed on other items), but C2 has to escape; hence, global fixed points are those with gcdc1,c2=1. For v9 there is dependence between the prey and the predator: the species C1 must eat C2 in order to survive, and C2 must escape from C1 in order to survive. Local fixed points are those with prime c2 and c2∉c1−1,c1+1 (cicadas-like behavior). Here, also F1c1,c2<0, so the species C1 disappears.Figure 8
For easier visual reference, here are canonical plots of three typical situations: gray denotes pairsc1,c2, c1,c2>1, such that gcdc1,c2=1, c2 is prime, or c2|c1, respectively.The first plot in each row shows the results of the simulation for the corresponding fitness tuple for the mutation interval−p,p=−1,1. The second plot in each row shows the results for the mutation interval −p,p=−4,4—as expected, by increasing mutation interval, some local fixed points cease to be fixed points, but global fitness points remain.We also simulated what happens in more “qualitative” setting, that is, when local fitness functions are generalized such that(16)fi:0,1×0,1⟶ℝ,i=1,2.In order to run these tests, each entry of fitness tuple was multiplied by a random integer from the interval1,10 and the mutation interval was −p,p=−1,1. The results are displayed in the third plot in each row. It is worth noting that the plots are indeed similar to the plots in the first two columns. These simulations indicate that our qualitative model covers well different qualitative cases, but this should by no means be considered a proof of such claim, without further analysis. Clearly, the analysis for any particular fitness tuple could be done if the specific case of interest appears.


## 4. Discussion

By analyzing the ecological factors that shape development, reproduction, and survival, the life-history theory seeks to explain the evolution of the major features of life cycles [21]. The complex spatial and temporal populational dynamics of periodical species are largely consequences of evolution of species interactions [22, 23]. Our models are relatively simple and describe accurately the evolution of several types of periodical species with synchronized emergence including species associated with prime-numbered (i.e., insects) and nonprime-numbered life cycles (i.e., plants) [24–26]. Several fitness tuples in our model show that various life cycles and their respective periodical emergences extinguish each other. On the other hand, our computer simulations demonstrate that the evolution of the periodicity is possible under restricted conditions. Here, we first comment four propositions and their respective classes of fitness tuples.In the classV1 (Proposition 1), there are two types of behavior. The tuples v7, v11 and v17 are of a “competition” type. In v17, the species C1 has a behavior like periodic cicadas with prime number fixed points (and in this case, the species C2 disappears, since the fitness F2c1,c2 is always negative). The other type of behavior is the one generated by the tuples v16 and v18. Here, one of the global fitness functions is constant (F2c1,c2=0 for v16 and F1c1,c2=1 for v18), while the other species has a bamboo-like behavior—it emerges as an integer multiple of the constant one. In both cases, there is a “commensalism” interaction, a special kind of symbiosis.The tuplesv1 and v2 from the class V2 (Proposition 2) generate two types of behavior. The tuple v1 generates symbiosis of the two species; that is, they have to appear at the same time. The tuple v2 generates a “commensalism” relation: the period c1 has to be an integer multiple of c2 (which is independent of c1), so it is also a kind of a “bamboo” effect.The tuplev6 (Proposition 3) is interesting since it is the case of two species that cannot emerge together—they are mutually harmful. In this sense, it is widely known that the competition exclusion principle can originate local extinction in populations [27] and could be represented by this proposition. According to the host-predator hypothesis, the parasitoids, predators, and certain microorganisms specializing on immature stages of a species lack prey or a substrate in the year when only adults are present. The following year, the predators are at low levels when the immature host numbers are again high. The same would hold for specialist predators of adult stage. Several mathematical models have been explored in an attempt to understand the origin of periodicity. Hoppensteadt and Keller [28] showed that synchronized emergence of a single year class is a possible consequence of predation given a limited environmental carrying capacity and life cycle lengths above a certain threshold value. Bulmer [29] also demonstrated that predation alone will not cause periodical behavior except as the accidental result of particular initial conditions. In this sense, our model is concordant with the last reported case.The tuplesv8 and v9 from the class V4 (Proposition 4) both generate predator-prey interaction, with the species C2 being the prey. For v8, C1 does not depend on C2 (it may feed on other items), but C2 has to escape, hence gcdc1,c2=1. For v9 there appears real dependence between the prey and the predator: the species C1 must eat C2 in order to survive, and C2 must escape from C1 in order to survive. In this case, classes of fixed points are those with prime c2 in which case F1c1,c2<0 so the species C1 disappears. This was the case studied widely in [20, 30].Some open questions and generalizations are related to considering full fitness tuples(17)v=f11,0,f20,1,f11,1,f21,1,f10,1,f21,0,f10,0,f20,0,and allowing values outside of the set −1,0,1. For example, large vegetative intervals of bamboos could be explained by the need to minimize energy [31], so, in this case, the “vegetative” values f10,1 or f21,0 might be positive. In particular, our simulation for the fitness tuple v=−1,1,0.8,1,1.25,0,0,0 generates fixed points with periods 26, 32, 34, 38, 39, and 40 for the species C1 (see Figure 9), which were observed in some species of long-period masting bamboos [2, 19]. Also, bamboos periodical life cycles can be dramatically affected during the flowering event by anthropogenic habitat modification that cause delayed reproduction [32]. That is, the combination of a fixed juvenile development time and a long adult life could favor the development of periodical behavior but does not guarantee it [33]. Again, this “qualitative” simulation is by no means proof or explanation of long-period masting; it is an example of the universality of our theoretical model.Figure 9
Simulation using full fitness tuple with values outside of the set−1,0,1 generates periods longer than 12 for both species, and the species C1 has periods 32, 34, 38, 39, and 40, as observed in some species of long-period masting bamboos.


## 5. Conclusion

The presented theoretical model covers all types of periodic behavior. We learned that, from a simple arithmetic point of view, the different interactions are related to only four principles. Given a couple of integer periodsc1,c2 the principles are as follows:(i)
gcdc1,c2=1 (predation or interspecific competition interaction as for tuples v6, v7, v8, v11 and v14)(ii)
One ofc1 or c2 has to be a prime (cicadas-like behavior as for v9, v10 and v17)(iii)
c1=c2 (symbiosis interaction as for v1 and v15)(iv)
One ofc1 or c2 has to be an integer multiple of the other (bamboos-like interaction as for v2, v16 and v18)Some of these principles are indeed observed in nature—periodical cicadas and long-period masting bamboos.To be periodical, a species must have a fixed life cycle length, and adults must appear synchronously, reproduce only once, and die [24]. We demonstrate that the emergence of synchrony in some species can occur by chance events, which disrupt this bet-hedging strategy and set the stage for periodicity. Our mathematical model predicts that, given certain initial conditions, intraspecific competition and predation favor its development. These synchronous species should be favored by natural selection later to establish the life cycles according to the interspecific interactions or environment restrictions. Finally, it is established that the variable intensity of interspecific competition, predation, and/or parasitism are interactions, which allow the existence of local stable positive periodical solutions for species such as insects and plants. However, as in any biological system, it is unlikely that one factor operates free from the influence of others.



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*Source: 2976351-2021-10-11.xml*