Linearly-perturbed MayLeonard model Dynamics of a linearly-perturbed MayLeonard competition model Gabriela Jaramillo1Lidia Mrad2and Tracy L. Stepien3

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Linearly-perturbed May–Leonard model

Dynamics of a linearly-perturbed May–Leonard competition model

Gabriela Jaramillo,1Lidia Mrad,2and Tracy L. Stepien3

1)Department of Mathematics, University of Houston, Houston, TX, 77204, USA

2)Department of Mathematics and Statistics, Mount Holyoke College, South Hadley, MA, 01075,

USA

3)Department of Mathematics, University of Florida, Gainesville, FL, 32611, USA

(*Electronic mail: gabriela@math.uh.edu ; lmrad@mtholyoke.edu ; tstepien@uﬂ.edu)

(Dated: 16 May 2023)

The May–Leonard model was introduced to examine the behavior of three competing populations where rich dynam-

ics, such as limit cycles and nonperiodic cyclic solutions, arise. In this work, we perturb the system by adding the

capability of global mutations, allowing one species to evolve to the other two in a linear manner. We ﬁnd that for

small mutation rates the perturbed system not only retains some of the dynamics seen in the classical model, such as

the three-species equal-population equilibrium bifurcating to a limit cycle, but also exhibits new behavior. For instance,

we capture curves of fold bifurcations where pairs of equilibria emerge and then coalesce. As a result, we uncover

parameter regimes with new types of stable ﬁxed points that are distinct from the single- and dual-population equilibria

characteristic of the original model. On the contrary, the linearly-perturbed system fails to maintain heteroclinic con-

nections that exist in the original system. In short, a linear perturbation proves to be signiﬁcant enough to substantially

inﬂuence the dynamics, even with small mutation rates.

Almost 50 years ago, May and Leonard1introduced an

extension of the classical Lotka–Volterra nonlinear sys-

tem to examine the long-term dynamics of three compet-

ing populations. In their work, they found that solutions

exhibit three distinct behaviors depending on the param-

eter values chosen, with the system approaching either a

stable ﬁxed point, a periodic orbit or, even more interest-

ingly, what is now known to be a heteroclinic cycle. In

the latter case, the observed trajectories are characterized

by nonperiodic oscillations of bounded amplitude but ever

increasing cycle time. Here, we establish and study an ex-

tended May–Leonard model by including a linear pertur-

bation that represents the ability of each species to adopt a

competing strategy. We ﬁnd that incorporating the linear

perturbation increases the number of physically-relevant

equilibrium states for certain parameter values. In addi-

tion, we also ﬁnd that the region in parameter space where

periodic orbits exist is much larger than in the case of the

original May–Leonard equations, and that the system no

longer exhibits nonperiodic cyclic solutions. Therefore,

allowing for a small linear mutation term representing

global mutations foments coexistence of different species.

In biological terms, this would imply that equipping popu-

lations with the possibility of switching from one strategy

to another with a small transition or mutation rate can fa-

vor biodiversity.

I. INTRODUCTION

During the last half a century, work on the May–Leonard

model1, a population dynamics model of three competing

species, and its variations has led to a variety of results. In par-

ticular, Schuster et al.2described the ω-limit set of the original

model and proved the existence of a heteroclinic cycle, while

Tang et al.3constructed a Lyapunov function to ﬁnd the basin

of attraction. It was also determined by Gaunersdorfer4that

the time averages of the trajectories tending to the heteroclinic

orbits in the model do not converge but spiral to the boundary

of a polygon. Approximate analytic solutions to the system

were also derived by Phillipson et al.5and conditions under

which the system is integrable have been studied by Leach

and Miritzis6, Llibre and Valls7, and Blé et al.8.

Extensions of the May–Leonard model1have included in-

corporating asymmetric competitive effects in order to deter-

mine conditions for existence and stability of limit cycles and

nonperiodic oscillations, as well as existence of ﬁrst integrals

of the Darboux type (Schuster et al.2, Chi et al.9, Wolkow-

icz10, Antonov et al.11,12). Instead of requiring equal intrinsic

growth rates for each competing population as in the May–

Leonard model1, existence of Hopf bifurcations and the sta-

bility of steady states were studied under the assumption of

unequal intrinsic growth rates (Coste et al.13, Zeeman14, van

der Hoff et al.15). Park16 extended the model to include an

external inﬂux and efﬂux of individuals into each population.

Balanced ﬂow among the groups resulted in persistent coex-

istence of all groups, including cases with oscillatory dynam-

ics, while imbalanced ﬂow resulted in various population sur-

vival states. More examples of various general three-species

competition models can be found in the review paper by Do-

bramysl et al.17.

The same year that the May–Leonard model1was pub-

lished, Gilpin18 considered the effects of adding a constant

perturbation to these equations. He found that this constant

term allowed for the formation of limit cycles in regions of pa-

rameter space where the original model exhibited only nonpe-

riodic oscillations. Other types of perturbations have not been

considered until more recently. For example, in 2014, Zhao

and Cen19 showed that adding small quadratic perturbations

to the model results in exactly one or two limit cycles bifur-

cating from the periodic orbits of the May–Leonard system.

Other perturbations recently studied have been periodic in na-

ture. In particular, it has been found that periodically forcing

arXiv:2210.04342v2 [math.DS] 12 May 2023

Linearly-perturbed May–Leonard model 2

the May–Leonard system results in the existence of strange at-

tractors (Rodrigues20). Additionally, periodic, quasiperiodic,

and chaotic solutions have been shown to exist under differ-

ent parameter conditions for small periodic perturbations to

the asymmetric May–Leonard model (Afraimovich et al.21)

and time-periodic perturbations to a general 3-D competitive

Lotka-Volterra model, of which the May–Leonard model is a

subcase (Chen et al.22).

In this work, we add a linear perturbation to the symmetric

May–Leonard model. This linear perturbation models muta-

tions among the competing populations, whereby individuals

in one class are able to mutate into another class. The ﬁrst to

examine these types of perturbations in a rock–paper–scissors

model with replicator-mutator equations was Mobilia23 about

a decade ago, followed by Toupo and Strogatz24, among oth-

ers (Yang et al.25, Park26, Hu et al.27, Mittal et al.28, Kabir

and Tanimoto29, Mukhopadhyay et al.30). Both the replicator

equations and the May–Leonard model have a similar struc-

ture, with the main difference being that in the former case the

unknowns represent fractions of a ﬁxed population, while in

the latter case the total population is not assumed to be a ﬁxed

number a priori.

As was the case in the May–Leonard equations, depending

on the parameter values chosen, the trajectories of the repli-

cator equations exhibit three types of long-term behavior. So-

lutions can either approach the equal-population stable ﬁxed

point, a heteroclinic cycle or, in contrast to the May–Leonard

model, one of the inﬁnitely many neutrally stable cycles that

ﬁll the state space. The effect of adding global mutations

to this system, where each species can mutate to any of the

other two with the same rate, is the loss of the saddle ﬁxed

points that form the heteroclinic cycle, as well as the emer-

gence of a stable limit cycle from a supercritical Hopf bifur-

cation for certain parameter values (Mobilia23). In contrast,

we ﬁnd that adding to the May–Leonard system a linear per-

turbation modeling global mutations increases the number of

physically-relevant steady states for certain parameter values,

and consequently changes the ensuing dynamics. We summa-

rize our ﬁndings, for small mutation rates, below:

• As expected, the perturbation changes the nature of

some steady states. We recover the trivial and equal-

population equilibria; however, we also ﬁnd a richer

variety of ﬁxed points that we view as perturbations of

the single- and dual-population equilibria found in the

May–Leonard model.

•

• In contrast to the May–Leonard system, the numerical

results we present suggest that the linearly-perturbed

May–Leonard model does not possess heteroclinic cy-

cles.

Practical applications of the May–Leonard model1in the

existing scientiﬁc literature are limited in number; to calcu-

late the cropping quotas for three competing herbivore species

(Fay and Greeff31). Such limited applicability stems, in part,

from the assumption that populations follow a cyclic domi-

nance competition pattern, which can be restrictive.

it is in aspect that the model’s equations resemble the repli-

cator equations used to model evolutionary games. Indeed,

evolutionary games are widely used in theoretical biology to

study interactions between species which follow a cyclic dom-

inance pattern (Mobilia23, Czárán et al.32, Kerr et al.33, Szol-

noki et al.34, Hofbauer and Sigmund35). Although this form

of competition seems to be rare in nature, there are a few ex-

amples where this behavior occurs. These include the mat-

ing strategies of side-blotched lizards (Sinervo and Lively36,

Zamudio and Sinervo37), and the interactions between three

different strains of E. coli (Kerr et al.33). In this context, mu-

tation can be seen as the ability of a population to change its

competing strategy. Previous work in this area by Toupo and

Strogatz24 and Mobilia23 has shown that global mutations re-

sult in the emergence of a limit cycle. It is perhaps then not

surprising that the numerical and analytical results we present

here lead to the same conclusion.

Outline: This paper is organized as follows. We ﬁrst re-

view the ﬁxed points of the May–Leonard model1and their

corresponding stability in Section II. We then introduce the

linearly-perturbed May–Leonard model in Section III, and ex-

plore how this modiﬁcation alters the dynamics for different

parameter regimes in Section IV. In particular, we ﬁnd that

the system fails to maintain the heteroclinic connections that

exist in the original model, justifying the lack of nonperiodic

oscillations we observe in simulations. Finally, we summarize

our ﬁndings in Section V, where we further comment on the

effects of adding the linear perturbation to the model.

II. MAY–LEONARD MODEL

May and Leonard1extended the classic Lotka–Volterra

equations for two competitors to a system of three competi-

tors, m1(t),m2(t), and m3(t), described by equations of the

general form

dmi(t)

dt =rimi(t) 1−

∑

j=1

αi jmj(t)!,i=1,2,3.(1)

Under symmetry assumptions that the intrinsic growth rates

are equal, r:=r1=r2=r3, and that the competitors affect

each other in a cyclic manner such that α:=α12 =α23 =

α31 and β:=α21 =α13 =α32, along with a rescaling of the

populations miand time tsuch that αii =1 and r=1, the

May–Leonard model1becomes

dm1

dt =m11−m1−αm2−βm3,(2a)

dm2

dt =m21−βm1−m2−αm3,(2b)

dm3

dt =m31−αm1−βm2−m3.(2c)

Solutions to (2) tend to one of the system’s 8 ﬁxed points, a

limit cycle, or a nonperiodic oscillation of bounded amplitude

but increasing cycle time.

Linearly-perturbed May–Leonard model 3

A. Fixed Points and Stability

The May–Leonard model (2) possesses 5 distinct nonnega-

tive ﬁxed points,

e0= (0,0,0),(3a)

e1= (1,0,0),e2= (0,1,0),e3= (0,0,1),(3b)

ec=1

1+α+β,1

1+α+β,(3c)

known to exist for all values of α,β>0, as well as 3 dual-

population ﬁxed points

f1=0,1−α

1−αβ ,1−β

1−αβ ,(4a)

f2=1−β

1−αβ ,0,1−α

1−αβ ,(4b)

f3=1−α

1−αβ ,1−β

1−αβ ,0,(4c)

for which positivity, and thus their physical relevance, de-

pends on the values of αand β. For example, for these ﬁxed

points to exist, we require αβ 6=1.

The stability of these ﬁxed points as a function of the two

parameters αand βis studied in depth in May and Leonard1.

Their results are summarized in the stability diagram in Fig. 1,

which we will also describe here.

(A)

Coexistence

(C)

Nonperiodic

Oscillation

(C)

Nonperiodic

Oscillation

Limit

Cycle

Limit

Cycle

0 1 2 3

FIG. 1. Stability diagram of ﬁxed points, limit cycles, and nonperi-

odic oscillations of the May–Leonard model (2). In Region A, only

the equal-population ﬁxed point ecis stable. In Region B, which is

bounded by the lines α=1 and β=1, the single-population ﬁxed

points e1,e2, and e3are stable. In Region C, nonperiodic oscillations

exist. Along the line α+β=2, limit cycles exist.

The ﬁxed point at the origin, e0, is always unstable. In

Region A, the only stable ﬁxed point is the equal-population

ﬁxed point, ec.

In Region B, the situation is reversed and all single-

population ﬁxed points, e1,e2, and e3, are stable, while the

ﬁxed point ecis now unstable. In this region, the long term

dynamics of the system depend on the initial conditions, and

thus the system approaches one of the ﬁxed points, e1,e2, or

e3, according to its initial conﬁguration.

In Region C, the system has nonperiodic cyclic solutions

that lie on the hyperplane m1+m2+m3=1. These solutions

approach and then leave each of the single-population ﬁxed

points. The time the system spends near each eiincreases

as the system evolves, and this loitering behavior follows a

logarithmic scale. On the border between Regions A and C,

where the parameters satisfy α+β=2, the system exhibits a

limit cycle.

III. MAY–LEONARD MODEL WITH LINEAR

PERTURBATIONS

We extend the May–Leonard model (1) to include linear

perturbations that are of the same form as the “global muta-

tions” in Toupo and Strogatz24, where each population mican

mutate into the other two with rate µ. The general form of this

linearly-perturbed May–Leonard model, is

dmi

dt =rimi 1−

∑

j=1

αi jmj!+µ



−2mi+

∑

j=1

j6=i

mj



,(5)

for i=1,2,3. Assuming, as in Section II, equal intrinsic

growth rates and that the competitors affect each other in a

cyclic manner, along with the same rescaling of populations

miand time t, (5) becomes

dm1

dt =m11−m1−αm2−βm3+µ−2m1+m2+m3,

(6a)

dm2

dt =m21−βm1−m2−αm3+µm1−2m2+m3,

(6b)

dm3

dt =m31−αm1−βm2−m3+µm1+m2−2m3.

(6c)

We assume that the competition parameters α,β>0 and

the mutation parameter µ>0. In the rest of this paper, we

study how the stability diagram of the May–Leonard model

(Fig. 1) changes when the mutation parameter, µ, in the

linearly-perturbed model (6) is nonzero.

IV. STABILITY DIAGRAM

In this section, we use perturbation analysis and the contin-

uation software package AUTO 0738 to investigate the effects

of the mutation parameter, µ, on the number and stability of

nonnegative ﬁxed points in system (6). Our results are sum-

marized in Fig. 2.

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Linearly-perturbedMayLeonardmodelDynamicsofalinearly-perturbedMayLeonardcompetitionmodelGabrielaJaramillo,1LidiaMrad,2andTracyL.Stepien31)DepartmentofMathematics,UniversityofHouston,Houston,TX,77204,USA2)DepartmentofMathematicsandStatistics,MountHolyokeCollege,SouthHadley,MA,01075,USA3)Departmento...

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Linearly-perturbed MayLeonard model Dynamics of a linearly-perturbed MayLeonard competition model Gabriela Jaramillo1Lidia Mrad2and Tracy L. Stepien3

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