Nonlinear eco-evolutionary games with global environmental fluctuations and
local environmental feedbacks

Yishen Jiang1,3, Xin Wang2,3,4,5,6,7*, Longzhao Liu2,3,4,5,6,7, Ming Wei1,3,
Jingwu Zhao8, Zhiming Zheng2,3,4,5,6,7,9,10, Shaoting Tang2,3,4,5,6,7,9,10*

1 School of Mathematical Sciences, Beihang University, Beijing 100191, China

2 Institute of Artificial Intelligence, Beihang University, Beijing 100191,
China

3 Key laboratory of Mathematics, Informatics and Behavioral Semantics (LMIB),
Beihang University, Beijing 100191, China

4 State Key Lab of Software Development Environment (NLSDE), Beihang
University, Beijing 100191, China

5 Zhongguancun Laboratory, Beijing, P.R.China

6 Beijing Advanced Innovation Center for Future Blockchain and Privacy
Computing, Beihang University, Beijing 100191, China

7 PengCheng Laboratory, Shenzhen 518055, China

8 School of Law, Beihang University, Beijing 100191, China

9 Institute of Medical Artificial Intelligence, Binzhou Medical University,
Yantai 264003, China

10 School of Mathematical Sciences, Dalian University of Technology, Dalian
116024, China

* wangxin_1993@buaa.edu.cn(XW); tangshaoting@buaa.edu.cn(ST)

## Abstract

Environmental changes play a critical role in determining the evolution of
social dilemmas in many natural or social systems. Generally, the
environmental changes include two prominent aspects: the global time-dependent
fluctuations and the local strategy-dependent feedbacks. However, the impacts
of these two types of environmental changes have only been studied separately,
a complete picture of the environmental effects exerted by the combination of
these two aspects remains unclear. Here we develop a theoretical framework
that integrates group strategic behaviors with their general dynamic
environments, where the global environmental fluctuations are associated with
a nonlinear factor in public goods game and the local environmental feedbacks
are described by the ‘eco-evolutionary game’. We show how the coupled dynamics
of local game-environment evolution differs in static and dynamic global
environments. In particular, we find the emergence of cyclic evolutions of
group cooperation and local environment, which forms an interior irregular
loop in the phase plane, depending on the relative changing speed of both
global and local environments compared to the strategic change. Our results
provide important insights toward how diverse evolutionary outcomes could
emerge from the nonlinear interactions between strategies and the changing
environments.

## Author summary

The intricate interplay between strategic behavior and environment is
ubiquitous in complex systems of different scales. Previous works mainly focus
on one aspect of the environmental changes: either global environment
fluctuations that unidirectionally decide the welfare of the evolutionary
dynamics, or local environment feedbacks that coevolve with the strategic
behavior. Here we develop a theoretical framework that integrates them both in
order to obtain a more complete picture of how group cooperation evolves in a
general dynamic environment. We show that global environmental fluctuations
can fundamentally alter the dynamical predictions of local game-environment
evolution. The most interesting finding is the emergence of cyclic evolutions
of group cooperation and local environment, which forms an interior irregular
loop in the phase plane, depending on the relative changing speed of both
global and local environments compared to the strategic change. Our results
show how rich dynamical outcomes arise from the interactions between strategic
behaviors and their natural or social environments, which has important
practical value for solving social dilemmas in an ever-changing world.

## Introduction

Cooperation promotes the emergence of stronger adaptabilities and more
abundant functions in many species, forming the very basis of natural systems
at different scales[1, 2, 3, 4]. However, the ‘selfish gene’ widely exists in
all biosystems where individuals always make rational choices based on their
own benefits[5, 6, 7], giving rise to social dilemmas of non-cooperation[8].
Naturally, understanding why the persistent cooperation occurs ubiquitously,
on what conditions a stable cooperation could be maintained or promoted, and
how the cooperative behavior evolves under natural selections has long been
the core objective of evolutionary game theory [9, 10, 11, 12]. In particular,
many different mechanisms have been proposed to address the well-known
Prisoner’s Dilemma and The Tragedy of the Commons [13, 14], for instance, kin
selection[15], direct reciprocity and indirect reciprocity[16, 17, 18],
punishment and reward [19, 20, 21], spatial reciprocity[22, 23, 24] and group
selection[25]. More realistically, the heterogeneities of the players are also
taken into account, such as network topology[26], selective participation
mechanism[10, 27], wealth-based selection [28], and the recently studied
higher-order interactions[29].

While early evolutionary game approach typically focuses on the internal
properties of replicator dynamics[9], assuming that the strategic interactions
happen in a fixed environment, the impact of a dynamic environment is ignored.
Accordingly, coevolutionary games that incorporate the evolution process of an
exogenous environment have been largely studied[10, 30]. Coevolution rules
introduce the environment-related characteristics into the game, for instance,
the interaction network, the size of population, the mobility, aging and
reputation of players, which also evolve in time and could affect the
evolutionary outcome of strategies[31]. Further, the ecological factors in
microbial systems are abstracted into a global time-dependent environment[32].
Such global environmental changes, reflecting the periodic ecological
fluctuations or the rapid ecological perturbations, modify the payoffs of the
game through a time-varying function and show highly complex impacts on the
evolution of group cooperation and on the evolutionary balance of
phenotypes[32, 33].

Note that coevolutionary games only consider the feedback from the global
environment, the dynamics of which is independent from the strategic
interactions. However, the bi-directional feedbacks between strategies and the
environment are identified in wide a range of real-world systems[34, 35, 36].
In microbial systems, cooperation often arises due to the secretion or the
release of extracellular enzymes, extracellular antibiotic compounds and etc,
which influences the local environmental state that will, in turn, alter the
incentive for public goods production[4, 2]. Likewise, in modern society,
decision-making dynamics of competitive cognitions can reshape the public
opinion environment, especially with the rise of large-scale social networks,
and this shared media atmosphere in turn affects the benefit of social
discussion in decision-making process[37]. In fact, the interactions between
online public discourse and external political environment can lead to the
emergence of polarized echo chambers[38], which has aroused great concern in
recent years[39]. Similar coupled dynamics can also be obtained in
psychological–economic systems and social-ecological systems[40, 36, 41],
relating to a number of big challenges such as global climate change,
overfishing, anti-vaccine problems, pandemic prevention and control[42, 43,
44, 45].

To characterize such complex feedback loops, an emerging theory of ‘eco-
evolutionary games’ is proposed recently[46, 47]. In eco-evolutionary game,
the strategic behaviors change the state of the environment, while in turn the
environment alters the payoff structure of the game, driving the replicator
dynamics with a strategy-dependent feedback-evolving game[48]. Abundant
evolutionary outcomes are observed under such framework. Beginning from the
simplest form, a two-player game coupled with linear environmental feedback
can already generate the persistent oscillation of both population
cooperations and environmental states[49]. Similar persistent cycles can also
occur in asymmetric games with heterogeneous environments[50, 51, 52]. As a
meaningful extension, a multi-player game coupled with asymmetrical
environmental feedback identifies the threshold of the feedback speed that can
yield oscillatory convergence to persistent cooperation, highlighting the
importance of time-scales[53]. An innovative manifold control approach is
further proposed to steer the eco-evolutionary dynamics to a desired
direction[54].

Despite the progress, current models consider only one aspect of the dynamic
environments, either global environmental changes which are time-dependent or
local environmental feedbacks which are strategy-dependent. In real-world
systems, however, these two aspects often coexist and exert complex forces on
the strategic evolutions. For example, in the context of the COVID-19
pandemic, cooperation in public health measures has strong impacts on disease
spreading, and vice versa[45]. Beyond this eco-evolution, the seasonal
fluctuations of virus’s transmissibility also alter the payoffs of the
strategic behaviors[55]. In crowdsourcing projects, cooperations can emerge
from the asymmetric incentive feedback which results in a local feedback
loop[53, 56]. Meanwhile, the periodic fluctuations of the global economic
environment[57], which obviously cannot be influenced by the strategic
behaviors within the projects, could affect the synergy and discounting of the
group payoffs[58]. Understanding the evolution of cooperation in such complex
systems thus has profound practical significance, calling for a framework that
could unveil the complete picture of the environmental effects where both
global and local environmental dynamics are incorporated.

Here we develop a theoretical framework that integrates group strategic
behaviors with their general dynamic environments. In specific, the global
environmental fluctuation that influences the group benefit is characterized
by a nonlinear factor in public goods game, while the local environmental
feedback driven by an asymmetric incentive mechanism is described by the ‘eco-
evolutionary game’. Of particular interest, we show how global environmental
fluctuations alter the dynamical predictions of the group cooperation in local
feedback-evolving game. Different from the previous observation that local
game-environment dynamics could eventually evolve to a stable interior fixed
point where cooperation and defection coexist, we find the emergence of cyclic
evolutions of group cooperation and local environment under a periodically
changing global environment. Surprisingly, such eco-evolutionary dynamics
could form an interior closed yet irregular orbit in the phase plane,
depending on the relative time-scale of both global and local environments
versus strategic changes. Our results provide novel insights toward how
complex group behaviors emerge from the nonlinear interactions between
strategies and dynamic environments, especially in a non-autonomous system,
which is important for understanding the evolution of social dilemmas in a
changing world.

## Materials and methods

Fig 1: Schematic of the eco-evolutionary games with general dynamic
environments. (Top) The group strategic behaviors are described by a nonlinear
evolutionary public goods game. (Top, Bottom) The influence of environmental
changes consist of two prominent aspects: the global environmental
fluctuations that directly affect the synergy and discounting of the group
payoffs, characterized by the nonlinear factor $w$, and the asymmetric
environmental feedbacks that drive the local strategy-dependent feedback-
evolving game, characterized by the multiplication factor of cooperators
$r_{c}$.

In order to study the evolution of group cooperation, which is ubiquitous in
microbial systems and in human society, we consider a modified nonlinear
public goods game (PGG) among a well-mixed infinitely large population. Each
of the $N$ participants can choose to be a cooperator contributing $c$ to the
public pool, or a defector reaping without sowing. In classic PGG, after the
total contribution is multiplied by the multiplication factor $r$, the total
benefit is equally distributed to each participant. However, it has been
obtained that group cooperation often emerges due to the the existence of
preferential access to the valuable common good or other extra incentives for
cooperators [53, 54], which brings about asymmetric payoff structures for
cooperators and defectors and further drives the local feedback-evolving game.
Here we distinguish multiplication factors of cooperators and defectors as
$r_{c}$ and $r_{d}$, respectively. In addition, the actual benefits provided
by cooperators may depend nonlinearly on the number of cooperators and on the
total investments, the former is common in biology and the later has been
largely revealed in economics [59]. Hence, we adopt the modeling idea proposed
in [58] and capture such nonlinearity by a nonlinear factor $w$.

Accordingly, the payoffs for each defector and cooperator in a group with $k$
cooperators, $P_{d}(k)$ and $P_{c}(k)$, are

$\displaystyle{P_{d}}\left(k\right)=\frac{{{r_{d}}c}}{N}\left({1+w+{w^{2}}+\cdots+{w^{k-1}}}\right)=\frac{{{r_{d}}c}}{N}\frac{{1-{w^{k}}}}{{1-w}},$
(1)
$\displaystyle{P_{c}}\left(k\right)=\frac{{{r_{c}}c}}{N}\left({1+w+{w^{2}}+\cdots+{w^{k-1}}}\right)-c=\frac{{{r_{c}}c}}{N}\frac{{1-{w^{k}}}}{{1-w}}-c,$
(2)

such that the benefits created by each additional cooperator are rescaled,
either synergistically enhanced when $w>1$, or discounted when $w<1$. Note
that when $w=1$, the classic linear PGG can be recovered. In a population with
a fraction $x$ of cooperators, for any focal individual, the probability that
$k$ out of $N-1$ other participants are cooperators is

$\displaystyle\left({\begin{array}[]{*{20}{c}}{N-1}\\\
k\end{array}}\right){x^{k}}{\left({1-x}\right)^{N-1-k}},$ (5)

The average fitness of defectors and cooperators, $f_{d}$ and $f_{c}$, are
thus given by

$\displaystyle{f_{d}}=\sum\limits_{k=0}^{N-1}{\left({\begin{array}[]{*{20}{c}}{N-1}\\\
k\end{array}}\right){x^{k}}{{\left({1-x}\right)}^{N-1-k}}{P_{d}}\left(k\right)}$
(8)
$\displaystyle=\frac{{{r_{d}}c}}{{N\left({1-w}\right)}}\left[{1-{{\left({1-x+wx}\right)}^{N-1}}}\right],$
(9)
$\displaystyle{f_{c}}=\sum\limits_{k=0}^{N-1}{\left({\begin{array}[]{*{20}{c}}{N-1}\\\
k\end{array}}\right){x^{k}}{{\left({1-x}\right)}^{N-1-k}}{P_{c}}\left({k+1}\right)}$
(12)
$\displaystyle=c\left\\{{\frac{{{r_{c}}}}{{N\left({1-w}\right)}}\left[{1-w{{\left({1-x+wx}\right)}^{N-1}}}\right]-1}\right\\}.$
(13)

For simplicity and without loss of generality, we specify that the
contribution of each cooperator is $1$. The changes in the fraction of
cooperation over time, namely the evolution of group cooperation, is then
described by the replicator dynamics:

$\displaystyle\dot{x}=x\left({{f_{c}}-\bar{f}}\right)=x\left({1-x}\right)\left({{f_{c}}-{f_{d}}}\right)$
$\displaystyle=x{\mkern 1.0mu}\left({1-x}\right){\mkern
1.0mu}\left({\frac{{{r_{c}}{\mkern 1.0mu}\left({w{\mkern
1.0mu}{{\left({w{\mkern 1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}-1-\frac{{{r_{d}}\left({{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}}\right),$ (14)

where $\bar{f}=x{f_{c}}+\left({1-x}\right){f_{d}}$ is the average fitness of
the population.

Further, we introduce two prominent aspects of the environmental influence
into the framework: the global environmental fluctuations and the local
environmental feedbacks. Generally, the global environment significantly
affects the total and marginal benefits of group cooperation. For instance,
the companies tend to increase the salary and recruit more employees in bull
markets, adopting aggressive expansion strategies as each additional staff
could provide much more returns, while in contrary, they are more likely to
cut salaries and reduce stuff in bear markets. Such global environmental
fluctuations could naturally be reflected and characterized by the nonlinear
factor $w$: a large synergy effect of cooperation corresponds to a good global
environment where $w>1$, while the discounting of cooperation is related to a
bad global environment where $w<1$. Therefore, we simply use a time-dependent
function $w=w\left(t\right)$ to depict changes of the global environment over
time.

In addition, the local environmental feedback arised from the asymmetric
incentive mechanism, which is strategy-dependent, is characterized by the
dynamics of cooperator’s multiplication factor $r_{c}$ following [53]:

$\displaystyle{\dot{r}_{c}}=\epsilon\left({{r_{c}}-\alpha}\right)\left({\beta-{r_{c}}}\right)f\left({x,{r_{c}}}\right),$
(15)

where $\epsilon>0$ denotes the relative changing speed of $r_{c}$ compared
with $x$. Due to limited resources in local environment, $r_{c}$ is confined
to the range $[\alpha,\beta]$ and we have $1<\alpha<\beta<N$ according to the
social dilemma in PGG. Moreover, $f\left({x,{r_{c}}}\right)$ describes the
asymmetric feedback mechanism in the model, whose sign determines the increase
or decrease in $r_{c}$. To mimic the fact that in social-economic systems such
as crowdsourcing projects, the authoritative organizer could distribute the
total payoffs for cooperators and defectors in an effort to promote
collaboration, we assume

$\displaystyle
f\left({x,{r_{c}}}\right)=-x{f_{c}}+\theta\left({1-x}\right){f_{d}},$ (16)

where $\theta>0$ denotes the distribution ratio of the expected total payoffs
of the cooperators and defectors. Such local feedback-evolving dynamics could
facilitate cooperation by increasing $r_{c}$ when $x$ is small, while in turn,
a relatively large $x$ leads to the decrease of cooperator’s rewards,
subjecting to the law of diminishing marginal utility. Besides, to reduce
model complexity, the defectors’ multiplication factor $r_{d}$ is set to be
constant and ${r_{d}}\leq\alpha$.

The complete modeling framework is illustrated by Fig. 1. The dynamics of our
nonlinear eco-evolutionary game with global environmental fluctuations and
local environmental feedbacks, which describes the complex group strategic
behaviors in general dynamic environments, can thus be written as

$\displaystyle\left\\{\begin{array}[]{l}\dot{x}={\mkern 1.0mu}x{\mkern
1.0mu}\left({1-x}\right){\mkern 1.0mu}\left({\frac{{{r_{c}}{\mkern
1.0mu}\left({w{\mkern 1.0mu}\left(t\right){{\left({w\left(t\right){\mkern
1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w\left(t\right)-1}\right)}}-1-\frac{{{r_{d}}\left({{{\left({w{\mkern
1.0mu}\left(t\right)x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w\left(t\right)-1}\right)}}}\right)\\\
{{\dot{r}}_{c}}=\epsilon\left({{r_{c}}-\alpha}\right){\mkern
1.0mu}\left({\beta-{r_{c}}}\right){\mkern 1.0mu}\left({-x{\mkern
1.0mu}\left({\frac{{{r_{c}}{\mkern 1.0mu}\left({w\left(t\right){\mkern
1.0mu}{{\left({w\left(t\right){\mkern
1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w\left(t\right)-1}\right)}}-1}\right)}\right.\\\
\left.{+\frac{{{r_{d}}\theta\left({1-x}\right){\mkern
1.0mu}\left({{{\left({w{\mkern
1.0mu}\left(t\right)x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w\left(t\right)-1}\right)}}}\right)}{\end{array}\right..$ (20)

## Results

### Nonlinear dynamics of local game-environment evolution in static global
environments

We first study the nonlinear dynamics of local game-environment evolution
under static global environments where $w(t)$ is a fixed constant.
Correspondingly, the eco-evolutionary game described by Eq. 20 degenerates to
an autonomous system. There are totally seven possible fixed points in the
system, six of which are on the boundary and the remaining one is an interior
equilibrium point (See detailed proof of the stability of all seven fixed
points in Appendix A). Only two boundary fixed points are possible to be
stable: (i) $\left(x^{*}=0\right)$ is always stable, leading to a full
defection among population which also occurs in the classic PGG. (ii)
$\left({{x^{*}}=1,r_{c}^{*}=\alpha}\right)$ is stable only if ${\mkern
1.0mu}\frac{{\alpha{\mkern 1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}>1$ and ${r_{d}}$ is smaller than the threshold
$r_{d}^{*}=\frac{{\alpha\left({{w^{N}}-1}\right)-N\left({w-1}\right)}}{{\left({{w^{N-1}}-1}\right)}}$.
Under this circumstance, the system evolves to the full cooperation with a
minimum value $\alpha$ of cooperators’ multiplication factor. The phase
diagram of the stability of this fixed point with respect to $w$ and $r_{d}$
is shown in Fig. 2(a). Not surprisingly, the full cooperation situation tends
to emerge from a healthy global environment with larger $w$ and smaller
$r_{d}$.

Fig 2: Effects of varying modeling parameters on equilibrium states of local
game-environment evolution in static global environments. Panel (a) shows the
combined influence of $w$ and $r_{d}$ on the stability of the boundary fixed
point $\left(1,\alpha\right)$. Panel (b) and (c) show the trends of the
interior fixed point $\left(x^{*},r_{c}^{*}\right)$ and the threshold of the
relative feedback speed $\epsilon^{*}$ as $\theta$ changes. In all panels,
$N=4$, $\alpha=1.5$, $\beta=3.5$.

Of particular interest, we analyze on what condition the interior equilibrium
point in the system

$\displaystyle({x^{*}}=\frac{\theta}{{\theta+1}},r_{c}^{*}=\frac{{N\left({w-1}\right)+{r_{d}}\left({{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}\right)}}{{w{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}}),$
(21)

could be stable, bringing about the coexistence of cooperators and defectors
with an intermediate local environmental state. As shown by Eq. 21, the final
frequency of cooperators $x^{*}$ is solely determined by $\theta$, the
distribution ratio of cooperator’s and defector’s total payoffs. As $\theta$
increases, $x^{*}$ also increases whereas the stable multiplication factor of
cooperators $r_{c}^{*}$ decreases (Fig. 2(b)). Such changing trends are
intuitive in many teamworks: an unskilled project, which has a relatively
larger $\theta$, often requires more participants with lower benefits. In
contrary, a skilled project such as scientific collaboration, the decrease of
$\theta$ results in small research teams with higher reward for each member.
Since $\alpha<r_{c}^{*}<\beta$, we first identify the existence condition of
this equilibrium point

$\displaystyle\max\left\\{{0,\frac{{\alpha\left({w{{\left({\frac{{w\theta+1}}{{\theta+1}}}\right)}^{N-1}}-1}\right)-N\left({w-1}\right)}}{{{{\left({\frac{{w\theta+1}}{{\theta+1}}}\right)}^{N-1}}-1}}}\right\\}\leq{r_{d}}$
$\displaystyle\leq\min\left\\{{\alpha,\frac{{\beta\left({w{{\left({\frac{{w\theta+1}}{{\theta+1}}}\right)}^{N-1}}-1}\right)-N\left({w-1}\right)}}{{{{\left({\frac{{w\theta+1}}{{\theta+1}}}\right)}^{N-1}}-1}}}\right\\}.$
(22)

Fig 3: Phase plane dynamics of local game-environment ($\bm{x-r_{c}}$)
evolution with $\bm{N=4}$, $\bm{\alpha=1.5}$, $\bm{\beta=3.5}$,
$\bm{\theta=2}$, $\bm{r_{d}=1}$. We set $w=0.8$, $1$, $1.2$ to describe the
scenarios of discounting, linear, and synergy PGG in each column, and the
corresponding $\epsilon^{*}$ are $0.2909$, $0.4444$, $1.0613$, respectively.
We therefore choose $\epsilon=0.0909,0.2444,0.8613$ for the first row and
$\epsilon=0.7909,0.9444,1.5613$ for the third row. The blue, pink and gray
areas represent the basin of attraction of different fixed points
$\left(x^{*}=0\right),\left(x^{*}=1,r_{c}^{*}=\alpha\right)$ and
(${x^{*}}=\frac{\theta}{{\theta+1}},r_{c}^{*}=\frac{{N\left({w-1}\right)+{r_{d}}\left({{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}\right)}}{{w{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}}$),
respectively.

Further, the existing interior fixed point is stable only if the relative
feedback speed of the local environment $\epsilon$ exceeds a threshold
$\epsilon^{*}$, which can be written as follows:

$\displaystyle{\epsilon^{*}}=\frac{{\left({r_{c}^{*}w-{r_{d}}}\right)\left({N-1}\right){{\left({w\theta+1}\right)}^{N-2}}\left({w-1}\right)}}{{\left({r_{c}^{*}-\alpha}\right)\left({\beta-
r_{c}^{*}}\right)\left[{w{{\left({w\theta+1}\right)}^{N-1}}-{{\left({1+\theta}\right)}^{N-1}}}\right]}}.$
(23)

Fig. 2(c) shows how $\epsilon^{*}$ varies as $w$, $r_{d}$ and $\theta$ change.
In a less optimistic global environment where the group benefits are
discounted ($w=0.8$), $\epsilon^{*}$ gently decreases as $\theta$ increases.
In a booming global environment where the group benefits are synergistically
enhanced ($w=1.2$), however, the trends become complicated, depending on the
multiplication factor of defectors $r_{d}$. When $r_{d}$ becomes smaller,
$\epsilon^{*}$ sharply increases as the interior fixed point tends to
disappear, the latter is shown by the dash line in Fig. 2(b).

In Fig. 3, we show phase dynamics of the local game-environment evolution
under different static global environments. We set $w=0.8$, $1$, $1.2$ to
mimic the scenarios of discounting, linear, and synergy PGG in each column,
respectively. Throughout the paper, we fix $N=4$, $\alpha=1.5$, $\beta=3.5$.
Other parameters are $\theta=2$ and $r_{d}=1$. Accordingly, we can calculate
the corresponding threshold $\epsilon^{*}$ using Eq. 23 and select parameters
$\epsilon$ that are smaller than, equal to or larger than $\epsilon^{*}$,
respectively. Consistent with our theoretical predictions, the persistent
coexistence of cooperators and defectors only occurs when
$\epsilon>\epsilon^{*}$ in all three scenarios, where the system oscillatorily
converges to the interior equilibrium state. In particular, the group
cooperation may only arise from the local asymmetrical environmental feedbacks
that are quick and timely enough, especially when the global environment is
relatively poor (Fig. 3(a)(d)(g) and Fig. 3(b)(e)(h)). Moreover, we find that
a good global environment significantly promotes the emergence of full
cooperation. An important insight is that a slower local environmental
feedback, though obstructs the emergence of interior equilibrium, could indeed
increase the basin of attraction of full cooperation (Fig. 3(c)(f)(i)). In
addition, as the benefits brought by the global environment $w$ increase, the
stable coexistence of cooperation and defection arise with less asymmetric
incentive feedback $r_{c}^{*}$, leading to a reduction in the basin of
attraction of the interior equilibrium. Our results show non-negligible role
and complicated joint influence of global environmental state and local
environmental feedback on group strategy evolutions.

### Nonlinear dynamics of local game-environment evolution in dynamic global
environments

Further, we study how dynamic global environments affect the local game-
environment evolution. Specifically, we focus on periodic global environmental
fluctuations, which is typical and widespread in various complex systems. For
instance, the daily cycle of sunlight, the seasonal fluctuations of ecological
characteristics, the cyclical economic crisis and etc. Here we consider two
types of periodic changes: discrete shifts modeled by a piecewise function
$w_{1}\left(t\right)$ and continuous fluctuations modeled by a continuous
function $w_{2}\left(t\right)$, which are simply given by

$\displaystyle{w_{\rm{1}}}\left(t\right)=\left\\{{\begin{array}[]{*{20}{c}}{1.2}&{\left[{\dfrac{t}{T}}\right]=2n}\\\
{0.8}&{\left[{\dfrac{t}{T}}\right]=2n+1}\end{array}}\right.,n=0,1,2,\cdots\cdots.$
(26)
$\displaystyle{w_{2}}\left(t\right)={1-0.5\sin\left({at+\delta}\right)}.$ (27)

where $a$ decides the time scales of global environmental fluctuations and
$\delta$ modulates the initial phase. Note that $w_{2}(t)$ is bounded in
$[0.5,1.5]$ in order to balance the degree of discounting and synergy. In
addition, for the convenience of comparison, we let the periods of
$w_{1}\left(t\right)$ and $w_{2}\left(t\right)$ be equal, i.e.,
$T=\frac{{\rm{\pi}}}{a}$.

Fig 4: Local game-environment evolution in a discretely varying global
environment. We use a periodic piecewise function, $w_{1}\left(t\right)$, to
describe the global environmental fluctuations. We uniformly select initial
points on the $x-r_{c}$ plane and plot the corresponding dynamical
trajectories by numerically solving Eq. 20. Trajectories that eventually
evolve to $\left(x^{*}=0\right)$, $\left(x^{*}=1,r_{c}^{*}=\alpha\right)$ or
circulate along an interior closed orbit are distinguished by blue, pink and
orange, respectively. In all panels, $N=4$, $\alpha=1.5$, $\beta=3.5$,
$\theta=1.2$, $r_{d}=0.7$.

In Fig. 4, we show the dynamical trajectories of local game-environment
evolutions under a discretely varying global environments
$w=w_{1}\left(t\right)$. The initial points are uniformly selected on the
phase plane and the trajectories are calculated numerically by Eq. 20. We fix
$\theta=1.2$ and $r_{d}=0.7$. The thresholds of the local environmental
feedback speed corresponding to the two values in $w_{1}\left(t\right)$ thus
can be obtained using Eq. 23, which are $\epsilon_{1}^{*}=0.5944$ and
$\epsilon_{2}^{*}=2.3578$. Therefore, we choose $\epsilon=0.5,2,5$, which
satisfies $0.5<\epsilon_{1}^{*}<2<\epsilon_{2}^{*}<5$, such that our analysis
could contain all the possible situations. Furthermore, considering the fact
that the global environmental changes are commonly much slower than the
strategy evolution, we set the relative changing speed $a=0.01$, $0.1$, $1$
and the corresponding periods are $2T=200\pi$, $20\pi$ and $2\pi$,
respectively. We find that the global environmental fluctuations could
fundamentally alter the dynamical predictions of the group cooperation in
local feedback-evolving game. When $\epsilon=0.5$, the local eco-evolutionary
dynamics evolves either to full defection where $x=0$ (blue trajectories) or
to full cooperation where $x=1$ and $r_{c}=\alpha$ (pink trajectories). When
$\epsilon=2$, however, a new evolutionary outcome emerges when the global
environment changes fastly: the local game-environment dynamics will
eventually circulate along an interior closed orbit (orange trajectories).
More specifically, different from the phenomenon that the system oscillatorily
converges to the interior equilibrium as we observed in static global
environments, here we find that a periodically changing global environment
could lead to the emergence of cyclic evolutions of group cooperation and
local environment. When $\epsilon=5$, such cyclic evolution can even emerge
from various time-scales of the global environmental changes.

Fig 5: Local game-environment evolution in a continuously changing global
environment. We use a periodic continuous function, $w_{2}\left(t\right)$, to
describe the global environmental fluctuations. Similarly, trajectories that
eventually evolve to $\left(x^{*}=0\right)$,
$\left(x^{*}=1,r_{c}^{*}=\alpha\right)$ or circulate along an interior closed
orbit are distinguished by blue, pink and orange, respectively. In all panels,
$N=4$, $\alpha=1.5$, $\beta=3.5$, $\theta=0.5$, $r_{d}=0.6$,
$\delta=-\frac{{\rm{\pi}}}{2}$.

Similar evolutionary trends can also be observed in a continuously changing
global environment $w=w_{2}\left(t\right)$ (Fig. 5). We set $\theta=0.5$,
$r_{d}=0.6$, $\delta=-\frac{{\rm{\pi}}}{2}$. The thresholds corresponding to
the maximum and minimum values of $w_{2}\left(t\right)$ are
$\epsilon_{max}=5.2864$ and $\epsilon_{min}=1.4528$. Thus we choose
$\epsilon=1,3,6$ such that $1<\epsilon_{min}<3<\epsilon_{max}<6$. Likewise,
the population will be occupied either by defectors or cooperators when
$\epsilon$ is small. When $\epsilon$ is at an intermediate value, the cyclic
evolution along an interior closed orbit arise from a rapidly changing global
environment. When $\epsilon$ is large, such interior closed orbit occurs at
all global time-scales. In addition, we confirm that the emergence of cyclic
evolution is robust on different multiplication factors of defector $r_{d}$
and on initial phase $\delta$ (see Appendix B). It is noteworthy that these
two factors have strong and complicated impacts on the range of initial values
that could eventually stabilize to full cooperation ($x^{*}=1$ and
$r_{c}^{*}=\alpha$) or to cyclic evolutions.

Fig 6: Emergence of cyclic evolutions of group cooperation and local
environment under periodically changing global environments given by
$\bm{w_{1}\left(t\right)}$ and $\bm{w_{2}\left(t\right)}$. The first column
shows local game-environment evolutions similar to Fig. 4 and Fig. 5. The
second column displays four typical dynamic trajectories in detail,
particularly the interior closed yet irregular orbits. The last two columns
present time evolutions of the frequency of cooperators $x$ and the
multiplication factor of cooperators $r_{c}$ under different initial
conditions, corresponding to the colored trajectories in the second column.
Parameters are $N=4$, $\alpha=1.5$, $\beta=3.5$, $\theta=1$, $r_{d}=1.2$,
$a=0.1$, $\epsilon=4$, $\delta=0$.

These results just show how diverse evolutionary group behaviors emerge from
the complex interactions between strategies and general dynamic environments,
especially highlight the important role of relative time-scales of both global
and local environments versus strategic changes ($a$ and $\epsilon$). Lastly,
we concentrate on the newly discovered phenomenon that the local game-
environment dynamics can evolve cyclically under different periodic global
environments (Fig. 6). Due to the complexity of non-autonomous system and the
lack of theoretical insight, we display in detail the phase plane dynamics of
local game-environment evolution, the typical dynamic trajectories with
different initial conditions, and the corresponding time evolutions of the
frequency of cooperators $x$ and the multiplication factor of cooperators
$r_{c}$, in order to provide a clear view of the cyclic evolutions. We set
$\theta=1$, $r_{d}=1.2$, $a=0.1$, $\delta=0$ and choose $\epsilon=4$ which is
larger than the maximum threshold of $\epsilon^{*}$ with regard to $w_{1}(t)$
and $w_{2}(t)$. Importantly, we find that the reason for the formation of
cyclic evolution under discrete global environment $w_{1}$ is that the two
fixed points are in the mutual attraction domains (Fig. 6(b)). In the shown
cases, the initial points first evolve to the stable fixed point
$\left({{x^{*}}=0.5,{r_{c}^{*}}=2.0047}\right)$ where $w=1.2$ within time $T$.
Subsequently, the global environment changes and fixed point $1$ becomes a new
initial point, which is captured by another stable fixed point
$\left({{x^{*}}=0.5,{r_{c}^{*}}=2.6996}\right)$ corresponding to $w=0.8$.
Similarly, in the next $T$ time, the evolution is directed to fixed point $1$
again, resulting in the formation of an interior closed orbit. The periodic
fluctuations of group cooperation and the local environment are shown in Fig.
6(c-d). In a continuously changing global environment described by $w_{2}(t)$,
however, the fixed point also varies over time which is unreachable for the
evolutionary process. Surprisingly, the interior closed yet irregular orbit
still emerges. Such periodic evolution is actually in line with the intuitions
from real world in a way that the group cooperation will neither disappear
completely nor always be maintained at the highest level in many complex
systems, for instance, the seasonal oscillating dynamics of the COVID-19
infection [60, 61, 62], and the dynamic adjustment of big companies in the
economic cycles. We interpret our results applying limit thought: the
evolution is always moving towards the nearest stable fixed point in each
small time interval, and the dynamical trajectories are approximately the
combinations of all such short paths connecting head to tail, the limit of
which finally converge to an closed yet irregular orbit as we observed. A
simple verification for the described thought using a discretely changing
global environment, which is characterized by a segmented function of four
segments, is provided in Appendix B.

## Conclusion and discussion

How rich dynamical outcomes arise from the interactions between strategic
behaviors and their natural or social environments is one of the fundamental
questions in many complex systems across disciplines. On one hand, the global
environment unidirectionally changes the total welfare of the evolutionary
dynamics. One the other hand, the local environment provides frequency-
dependent feedback that modifies the payoff structure of the game dynamics,
while in turn the strategies taken by individuals can also reshape the state
of the local environment over time. Particularly, two important theoretical
frameworks that describe the latter feedback loop are proposed: the stochastic
game[63, 64], and the eco-evolutionary game[49, 52, 53, 54]. Stochastic games
introduce game transition mechanisms to depict the discrete changes of the
external environments, i.e., the cooperation behavior in the current game can
affect the game that individuals play in the next period. Furthermore, eco-
evolutionary game theory characterizes the continuous environmental changes
coupled with strategic interactions via a set of ordinary differential
equations. However, previous models exclusively focus on one aspect of the
environmental changes. To obtain a more complete picture of the group
behavioral evolutions in a general dynamic environment, we develop a modeling
framework that integrates the more complicated influence exerted by both
global environmental fluctuations and local environmental feedbacks.

Real interactions between strategic behaviors and their environments are
commonly nonlinear. Our analysis shows how this nonlinearity, relating to the
state of the global environment and the marginal benefits provided by the
cooperators, affect the local game-environment evolution. We find that in a
static global environment, regardless of the scenarios of discounting, linear,
and synergy, the persistent coexistence of cooperation and defection only
emerges if the relative feedback speed of the local environment exceeds a
certain threshold, breaking the ‘Tragedy of the Commons’. The nonlinear
factor, however, could determine the occurrence of full cooperation and
influence the attraction basin of the stable interior equilibrium.

As the influence of dynamic global environments on local eco-evolutionary game
is the primary focus of our model, our results emphatically show how the
periodic global environmental fluctuations fundamentally alter the
evolutionary outcomes of the group cooperation. The most intriguing finding is
the emergence of an interior closed yet irregular orbit in local game-
environment phase plane, leading to cyclic evolutions of group cooperation and
local environment, which is firstly discovered in multi-player situation and
qualitatively in line with the oscillating dynamics in two-player games [49,
52]. Further, we unveil that this new dynamical phenomenon can be intuitively
understood as a limit of continuously converging process of the dynamical
paths confined by the fixed points. Last and most important, our theoretical
framework highlight the crucial role of relative time-scales of both global
and local environments compared to strategic interactions.

To sum up, our model provides profound insight into how diverse group
behaviors, especially oscillating convergence and cyclic evolutions, can
emerge from the nonlinear interactions between strategies and dynamic
environments, which has important practical value for solving social dilemmas
in an ever-changing world. For instance, in social-economic system, the
companies engaging staff can be mapped into the local environment, the state
of which coevolves with individual strategies[65]. Outside, the trends of the
corresponding industry, which can largely determine the marginal benefits of
strategic behaviors, can be modeled as the global environment. Our framework
also has great potential in studying more complicated cases as well as bigger
issues in human society, such as the coevolution of legal environment, new
technologies and the economics[66].

Though in this work we only consider the periodic fluctuations of the global
environment which are widespread in natural systems, various changing rules
can be studied by simply giving different time-evolving functions $w(t)$.
Future works may further consider the structured interactions with increasing
complexity, and the control strategies in the framework of game-environment
dynamics.

## Acknowledgments

This work is supported by Program of National Natural Science Foundation of
China Grant No. 12201026,11871004, 11922102, 62141605 and National Key
Research and Development Program of China Grant No. 2018AAA0101100,
2021YFB2700304.

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## Appendix A. Stability of fixed points in static global environments

For any given $w$ ($w\neq 1$), the dynamic equations of the eco-evolutionary
system can be written as

$\displaystyle\left\\{\begin{array}[]{l}\dot{x}={\mkern 1.0mu}x{\mkern
1.0mu}\left({1-x}\right){\mkern 1.0mu}\left({\frac{{{r_{c}}{\mkern
1.0mu}\left({w{\mkern 1.0mu}{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}-1-\frac{{{r_{d}}\left({{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}}\right)\\\
{{\dot{r}}_{c}}=\epsilon\left({{r_{c}}-\alpha}\right){\mkern
1.0mu}\left({\beta-{r_{c}}}\right){\mkern 1.0mu}\left({-x{\mkern
1.0mu}\left({\frac{{{r_{c}}{\mkern 1.0mu}\left({w{\mkern
1.0mu}{{\left({w{\mkern 1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}-1}\right)}\right.\\\
\left.{+\frac{{{r_{d}}\theta\left({1-x}\right){\mkern
1.0mu}\left({{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}}\right)}{\end{array}\right..$ (31)

Jacobian matrix of this system is

$J=\left[{\begin{array}[]{*{20}{c}}\begin{array}[]{l}A\\\
C\end{array}&\begin{array}[]{l}B\\\ D\end{array}\end{array}}\right],$ (32)

where

$\displaystyle A=\left({2x-1}\right){\mkern
1.0mu}\left({\frac{{{r_{d}}\left({{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}-\frac{{{r_{c}}{\mkern 1.0mu}\left({w{\mkern
1.0mu}{{\left({w{\mkern 1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}+1}\right)$ $\displaystyle+\frac{{{\mkern
1.0mu}x\left({1-x}\right){\mkern 1.0mu}\left({{r_{c}}w-{r_{d}}}\right){\mkern
1.0mu}\left({N-1}\right){\mkern 1.0mu}{\mkern 1.0mu}{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-2}}}}{N}$ $\displaystyle
B=\frac{{x\left({1-x}\right)\left({w{\mkern 1.0mu}{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-1}}-1}\right){\mkern 1.0mu}}}{{N{\mkern
1.0mu}\left({w-1}\right)}}$ $\displaystyle C=\epsilon{\mkern
1.0mu}\left({{r_{c}}-\alpha}\right){\mkern
1.0mu}\left({\beta-{r_{c}}}\right){\mkern
1.0mu}\left({\frac{{{r_{c}}\left({1-w{\mkern 1.0mu}{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-1}}}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}}\right.$
$\displaystyle\left.{+\frac{{{r_{d}}\theta{\mkern
1.0mu}\left({1-{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-1}}}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}+\frac{{{r_{d}}\theta{\mkern
1.0mu}\left({N-1}\right){\mkern 1.0mu}\left({1-x}\right){\mkern
1.0mu}{{\left({w{\mkern 1.0mu}x-x+1}\right)}^{N-2}}}}{N}}\right.$
$\displaystyle\left.{-\frac{{{r_{c}}w{\mkern 1.0mu}x{\mkern
1.0mu}\left({N-1}\right){\mkern 1.0mu}{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-2}}}}{N}+1}\right)$ $\displaystyle
D=\epsilon\left({2{r_{c}}-\alpha-\beta}\right){\mkern 1.0mu}\left({x{\mkern
1.0mu}\left({\frac{{{r_{c}}{\mkern 1.0mu}\left({w{\mkern
1.0mu}{{\left({w{\mkern 1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}-1}\right)}\right.$
$\displaystyle\left.{+\frac{{{r_{d}}\theta{\mkern
1.0mu}\left({x-1}\right){\mkern 1.0mu}\left({{{\left({w{\mkern
1.0mu}x-x+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}}\right){\mkern 1.0mu}$
$\displaystyle+\frac{{\epsilon{\mkern 1.0mu}x{\mkern 1.0mu}\left({1-w{\mkern
1.0mu}{{\left({w{\mkern 1.0mu}x-x+1}\right)}^{N-1}}}\right){\mkern
1.0mu}\left({{r_{c}}-\alpha}\right){\mkern
1.0mu}\left({\beta-{r_{c}}}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}$
(33)

Solving $\dot{x}=0$ and ${\dot{r}_{c}=0}$, we can derive the following fixed
points.

(1) ${x^{*}}=0$

$\displaystyle
J\left({{x^{*}}=0,{r_{c}}}\right)=\left[{\begin{array}[]{*{20}{c}}{\frac{{{r_{c}}}}{{{\mkern
1.0mu}N}}-1}&0\\\ {\epsilon{\mkern
1.0mu}\left({{r_{c}}-\frac{3}{2}}\right){\mkern
1.0mu}\left({\frac{{\rm{7}}}{{\rm{2}}}-{r_{c}}}\right){\mkern
1.0mu}\left({\frac{{{\mkern
1.0mu}{r_{d}}\theta\left({N-1}\right)}}{N}-\frac{{{r_{c}}}}{N}+1}\right)}&0\end{array}}\right]$
(34)

Eigenvalues are
${\lambda_{\rm{1}}}{\rm{=}}\frac{{{r_{c}}-\left({N+1}\right)}}{N}<0$ (since
${r_{c}}\leq\alpha<N$), ${\lambda_{2}}=0$. So this fixed point is stable.

(2) ${x^{*}}=1,r_{c}^{*}=\alpha$

$\displaystyle
J\left({1,\alpha}\right)=\left({\begin{array}[]{*{20}{c}}{\frac{{{r_{d}}\left({{w^{N-1}}-1}\right)-\alpha\left({{w^{N}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}+1}&0\\\ 0&{\epsilon{\mkern
1.0mu}\left({\beta-\alpha}\right){\mkern 1.0mu}\left({1-\frac{{\alpha{\mkern
1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}}\right)}\end{array}}\right)$ (35)

Eigenvalues are
${\lambda_{1}}=\frac{{{r_{d}}\left({{w^{N-1}}-1}\right)-\alpha\left({{w^{N}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}+1$ and ${\lambda_{2}}=\epsilon{\mkern
1.0mu}\left({\beta-\alpha}\right){\mkern 1.0mu}\left({1-\frac{{\alpha{\mkern
1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}}\right)$. When ${\mkern 1.0mu}\frac{{\alpha{\mkern
1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}>1$ and
${r_{d}}<\frac{{\alpha\sum\limits_{k=0}^{N-1}{{w^{k}}}-N}}{{\sum\limits_{k=0}^{N-2}{{w^{k}}}}}=\frac{{\alpha\left({{w^{N}}-1}\right)-N\left({w-1}\right)}}{{\left({{w^{N-1}}-1}\right)}}=r_{d}^{*}$,
we have ${\lambda_{1}}<0$ and ${\lambda_{2}}<0$ and this fixed point is
stable.

(3) ${x^{*}}=1,r_{c}^{*}=\beta$

$\displaystyle
J\left({1,\beta}\right){\rm{=}}\left({\begin{array}[]{*{20}{c}}{\frac{{{r_{d}}\left({{w^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}-\frac{{\beta{\mkern
1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}+1}&0\\\
0&{\epsilon{\mkern 1.0mu}\left({\beta-\alpha}\right){\mkern
1.0mu}\left({\frac{{\beta{\mkern 1.0mu}\left({{\mkern
1.0mu}{w^{N}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}-1}\right)}\end{array}}\right)$ (36)

Eigenvalues are
${\lambda_{1}}=\frac{{{r_{d}}\left({{w^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}-\frac{{\beta{\mkern
1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}+1$ and
${\lambda_{2}}=\epsilon{\mkern 1.0mu}\left({\beta-\alpha}\right){\mkern
1.0mu}\left({\frac{{\beta{\mkern 1.0mu}\left({{\mkern
1.0mu}{w^{N}}-1}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}-1}\right)$. We
can conclude that it is impossible for ${\lambda_{1}}$ and ${\lambda_{2}}$ to
be negative simultaneously, the proof of which is as follows: If
${\lambda_{2}}<0$, we have $\frac{{\beta{\mkern 1.0mu}\left({{\mkern
1.0mu}{w^{N}}-1}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}<1$.
Furthermore, if ${\lambda_{1}}<0$, we have
$\frac{{{r_{d}}\left({{w^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}<\frac{{\beta{\mkern
1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}-1<0$.
Then we have ${r_{d}}<0$, which is contradicted with the real situation
${r_{d}}\geq 0$. Thus, there is at least one eigenvalue no less than $0$,
which indicates that the fixed point is unstable.

(4) ${x^{*}}=1,r_{c}^{*}=\frac{{N\left({w-1}\right)}}{{{w^{N}}-1}}$ (If
$\alpha<r_{c}^{*}<\beta$, the fixed point is in domain of definition)

$\displaystyle
J\left({{x^{*}},r_{c}^{*}}\right)=\left[{\begin{array}[]{*{20}{c}}A&0\\\
C&D\end{array}}\right]$ (37)

where

$\displaystyle A$ $\displaystyle=\frac{{{r_{d}}{\mkern
1.0mu}\left({{w^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}-\frac{{r_{c}^{*}{\mkern
1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}+1=\frac{{{r_{d}}{\mkern
1.0mu}\left({{w^{N-1}}-1}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}>0$
(38) $\displaystyle C$ $\displaystyle=\epsilon{\mkern
1.0mu}\left({\alpha-{r_{c}}}\right){\mkern
1.0mu}\left({\beta-{r_{c}}}\right){\mkern 1.0mu}\left({\frac{{r_{c}^{*}{\mkern
1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}+\frac{{{r_{d}}\theta{\mkern
1.0mu}\left({{w^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}+\frac{{r_{c}^{*}{w^{N-1}}{\mkern
1.0mu}\left({N-1}\right)}}{N}-1}\right)$ $\displaystyle D$
$\displaystyle=\frac{{\epsilon\left({\alpha-r_{c}^{*}}\right){\mkern
1.0mu}\left({\beta-r_{c}^{*}}\right){\mkern
1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}<0$

Eigenvalues are ${\lambda_{1}}=\frac{{{r_{d}}{\mkern
1.0mu}\left({{w^{N-1}}-1}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}>0$ and
${\lambda_{2}}=\frac{{\epsilon\left({\alpha-r_{c}^{*}}\right){\mkern
1.0mu}\left({\beta-r_{c}^{*}}\right){\mkern
1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern 1.0mu}\left({w-1}\right)}}<0$. This
fixed point is a saddle point.

(5) ${x^{*}}=\frac{{{{\left({\frac{{\alpha-{r_{d}}+N{\mkern
1.0mu}\left({w-1}\right)}}{{\alpha
w-{r_{d}}}}}\right)}^{\frac{{\rm{1}}}{{N-1}}}}-1}}{{w-1}},r_{c}^{*}=\alpha$
(If $0<{x^{*}}<1$, the fixed point is in domain of definition)

$\displaystyle
J\left({{x^{*}},\alpha}\right)=\left[{\begin{array}[]{*{20}{c}}A&B\\\
0&D\end{array}}\right]$ (39)

where

$\displaystyle A=\frac{{{x^{*}}{\mkern
1.0mu}\left({1-{x^{*}}}\right)\left({\alpha{\mkern
1.0mu}w-{r_{d}}}\right){\mkern 1.0mu}\left({N-1}\right){\mkern 1.0mu}{\mkern
1.0mu}{{\left({w{\mkern 1.0mu}{x^{*}}-{x^{*}}+1}\right)}^{N-2}}}}{N}$
$\displaystyle B=\frac{{{x^{*}}\left({1-{x^{*}}}\right){\mkern
1.0mu}\left({w{\mkern 1.0mu}{{\left({w{\mkern
1.0mu}{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}\right){\mkern 1.0mu}}}{{N{\mkern
1.0mu}\left({w-1}\right)}}$ $\displaystyle D=\epsilon{\mkern
1.0mu}\left({\beta-\alpha}\right){\mkern 1.0mu}\left({-{x^{*}}{\mkern
1.0mu}\left({\frac{{\alpha\left({w{\mkern 1.0mu}{{\left({w{\mkern
1.0mu}{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}-1}\right)}\right.$
$\displaystyle\left.{+\frac{{{\mkern 1.0mu}{r_{d}}\theta{\mkern
1.0mu}\left({1-{x^{*}}}\right){\mkern 1.0mu}\left({{{\left({w{\mkern
1.0mu}{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}}\right)$ (40)

Eigenvalues are ${\lambda_{1}}=A$ and ${\lambda_{2}}=D$. Here we consider the
two following situations: (1) $w>1$ and (2) $w<1$.

Case $w>1$: Due to ${r_{d}}<\alpha<w\alpha$, we have $\alpha w-{r_{d}}>0$,
which indicates ${\lambda_{1}}>0$. Therefore, this fixed point is unstable.

Case $w<1$: Assume ${\lambda_{1}}<0$. This assumption leads to $\alpha
w-{r_{d}}<0$. Here we set $N=4$. Considering
$0<{x^{*}}=\frac{{{{\left({\frac{{\alpha-{r_{d}}+N{\mkern
1.0mu}\left({w-1}\right)}}{{\alpha
w-{r_{d}}}}}\right)}^{\frac{{\rm{1}}}{{N-1}}}}-1}}{{w-1}}<1$, we can derive
${w^{3}}<\frac{{\alpha-{r_{d}}+4{\mkern 1.0mu}\left({w-1}\right)}}{{\alpha
w-{r_{d}}}}<1$. By calculating the right half of the inequality, we have
$\alpha-{r_{d}}+4\left({w-1}\right)>\alpha w-{r_{d}}$, i.e. $w>1$.
Contradictory appears, so the assumption does not hold, which means that this
fixed point is unstable.

(6) ${x^{*}}=\frac{{{{\left({\frac{{\beta-{r_{d}}+N{\mkern
1.0mu}\left({w-1}\right)}}{{\beta
w-{r_{d}}}}}\right)}^{\frac{{\rm{1}}}{{N-1}}}}-1}}{{w-1}},r_{c}^{*}=\beta$ (If
$0<{x^{*}}<1$, the fixed point is in domain of definition)

Jacobian is similar to (5), one of whose eigenvalues is
$\lambda_{1}=A=\frac{{{x^{*}}{\mkern
1.0mu}\left({1-{x^{*}}}\right)\left({\beta w-{r_{d}}}\right){\mkern
1.0mu}\left({N-1}\right){\mkern 1.0mu}{\mkern 1.0mu}{{\left({w{\mkern
1.0mu}{x^{*}}-{x^{*}}+1}\right)}^{N-2}}}}{N}>0$. Thus, this fixed point is
unstable.

(7) Interior fixed point
$\left({{x^{*}}=\frac{\theta}{{\theta+1}},r_{c}^{*}=\frac{{N\left({w-1}\right)+{r_{d}}\left({{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}\right)}}{{w{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}}}\right)$
(If $\alpha\leq r_{c}^{*}\leq\beta$, the fixed point is in domain of
definition)

$\displaystyle
J\left({{x^{*}},r_{c}^{*}}\right)=\left[{\begin{array}[]{*{20}{c}}A&B\\\
C&D\end{array}}\right]$ (41)

where

$\displaystyle A=\frac{{{\mkern 1.0mu}{x^{*}}\left({1-{x^{*}}}\right){\mkern
1.0mu}\left({r_{c}^{*}w-{r_{d}}}\right){\mkern 1.0mu}\left({N-1}\right){\mkern
1.0mu}{\mkern 1.0mu}{{\left({w{\mkern
1.0mu}{x^{*}}-{x^{*}}+1}\right)}^{N-2}}}}{N}$ $\displaystyle
B=\frac{{{x^{\rm{*}}}\left({1-{x^{\rm{*}}}}\right)\left({w{\mkern
1.0mu}{{\left({w{\mkern
1.0mu}{x^{\rm{*}}}-{x^{\rm{*}}}+1}\right)}^{N-1}}-1}\right){\mkern
1.0mu}}}{{N{\mkern 1.0mu}\left({w-1}\right)}}$ $\displaystyle
C=-{f_{c}}\left({{x^{*}},r_{c}^{*}}\right)-\theta{f_{d}}\left({{x^{*}},r_{c}^{*}}\right)-xf_{c}^{{}^{\prime}}\left({{x^{*}},r_{c}^{*}}\right)+\theta\left({1-x}\right)f_{d}^{{}^{\prime}}\left({{x^{*}},r_{c}^{*}}\right)$
$\displaystyle=-\left({1+\theta}\right){f_{d}}\left({{x^{*}},r_{c}^{*}}\right)-xf_{c}^{{}^{\prime}}\left({{x^{*}},r_{c}^{*}}\right)+\theta\left({1-x}\right)f_{d}^{{}^{\prime}}\left({{x^{*}},r_{c}^{*}}\right)$
$\displaystyle={\mkern
1.0mu}\epsilon\left({r_{c}^{*}-\alpha}\right)\left({\beta-
r_{c}^{*}}\right)\left({\frac{{\left({1+\theta}\right){r_{d}}\left({1-{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}}\right)}}{{N\left({w-1}\right)}}}\right.$
$\displaystyle\left.{+\left[{{r_{d}}\theta\left({1-{x^{*}}}\right)-r_{c}^{*}w{x^{*}}}\right]\frac{{\left({N-1}\right){{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-2}}}}{N}}\right)$
$\displaystyle D={\mkern 1.0mu}\epsilon{\mkern
1.0mu}\left({r_{c}^{*}-\alpha}\right){\mkern 1.0mu}\left({\beta-
r_{c}^{*}}\right)\frac{{{x^{*}}{\mkern 1.0mu}\left({1-w{\mkern
1.0mu}{{\left({w{\mkern 1.0mu}{x^{*}}-{x^{*}}+1}\right)}^{N-1}}}\right){\mkern
1.0mu}}}{{N{\mkern 1.0mu}\left({w-1}\right)}}$ (42)

This fixed point is stable if and only if the following two conditions holds:

$\displaystyle\left\\{\begin{array}[]{l}AD-BC>0\\\ A+D<0\end{array}\right.$
(43)

The first equality $AD-
BC=\frac{{\epsilon\left({r_{c}^{*}-\alpha}\right)\left({\beta-
r_{c}^{*}}\right){x^{*}}\left({1-{x^{*}}}\right)}}{{{N^{2}}}}\frac{{w{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}}{{w-1}}\frac{{{r_{d}}\left({1+\theta}\right)\left({{{\left({w{x^{\rm{*}}}-{x^{\rm{*}}}+1}\right)}^{N-1}}-1}\right)}}{{w-1}}>0$
always hold. Thus this point is stable if and only if $A+D>0$, which is
equivalent to the following condition:

$\displaystyle\epsilon>{\epsilon^{*}}$ (44)

where

$\displaystyle{\epsilon^{*}}$
$\displaystyle=\frac{{\left({1-{x^{*}}}\right)\left({r_{c}^{*}w-{r_{d}}}\right)\left({N-1}\right){{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-2}}\left({w-1}\right)}}{{\left({r_{c}^{*}-\alpha}\right)\left({\beta-
r_{c}^{*}}\right)\left[{w{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}\right]}}$
(45)
$\displaystyle=\frac{{\left({r_{c}^{*}w-{r_{d}}}\right)\left({N-1}\right){{\left({w\theta+1}\right)}^{N-2}}\left({w-1}\right)}}{{\left({r_{c}^{*}-\alpha}\right)\left({\beta-
r_{c}^{*}}\right)\left[{w{{\left({w\theta+1}\right)}^{N-1}}-{{\left({1+\theta}\right)}^{N-1}}}\right]}}$

In conclusion, on the premise of $w\neq 1$, the stable fixed points and the
conditions of stability are as follows:

(1) The fixed points on the vertical-axes $\left(x^{*}=0\right)$ are always
stable.

(2) The fixed point $\left({x^{*}}=1,r_{c}^{*}=\alpha\right)$ is stable when
${\mkern 1.0mu}\frac{{\alpha{\mkern 1.0mu}\left({{w^{N}}-1}\right)}}{{N{\mkern
1.0mu}\left({w-1}\right)}}>1$ and
${r_{d}}<\frac{{\alpha\left({{w^{N}}-1}\right)-N\left({w-1}\right)}}{{\left({{w^{N-1}}-1}\right)}}$.

(3) The interior fixed point
$\left({{x^{*}}=\frac{\theta}{{\theta+1}},r_{c}^{*}=\frac{{N\left({w-1}\right)+{r_{d}}\left({{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}\right)}}{{w{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}}}\right)$
is stable when
$\epsilon>\frac{{\left({1-{x^{*}}}\right)\left({r_{c}^{*}w-{r_{d}}}\right)\left({N-1}\right){{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-2}}\left({w-1}\right)}}{{\left({r_{c}^{*}-\alpha}\right)\left({\beta-
r_{c}^{*}}\right)\left[{w{{\left({w{x^{*}}-{x^{*}}+1}\right)}^{N-1}}-1}\right]}}$.

## Appendix B. The influence of multiplication factors of defectors $r_{d}$
and initial phase $\delta$.

Fig 7: Game-environment evolution under different multiplication factors of
defector $\bm{r_{d}}$. $x-r_{c}$ phase plane shows that trajectories
eventually evolve to $\left(x^{*}=0\right)$,
$\left(x^{*}=1,r_{c}^{*}=\alpha\right)$ or circulate along an interior closed
orbit, which are distinguished by blue, pink and orange, respectively. In all
panels, $N=4$, $\alpha=1.5$, $\beta=3.5$, $\theta=0.5$, $a=1$, $\epsilon=6$,
$\delta=0$.

In this part, we mainly aim to explore how multiplication factors of defectors
$r_{d}$ and initial phase $\delta$ affect the emergence of the interior closed
orbit.

We first study the influence of $r_{d}$. Figure 7 presents the phase plane of
game-environment evolution showing trajectories of different initial points.
Specifically, Figure 7(a)(b) describes situations in discretely varying global
environment under different $r_{d}$, while Figure 7(c)(d) corresponds to
continuously changing global environment. In each figure, there are always
three kinds of trajectories represented by different colors, where orange
lines represent trajectories evolving along an interior closed orbit. It
indicates that the emergence of the interior closed orbit is robust on
different $r_{d}$, whatever the environment is discretely varying or
continuously changing. In addition, by comparing Figure 7(a) and (b) or Figure
7(c) and (b), we find that the value of $r_{d}$ indeed affect the range of
orange region, i.e., the range of trajectories eventually evolving along the
interior closed orbit.

Then, we explore the influence of $\delta$, the initial phase of the function
$w_{2}(t)$ representing the continuously changing environment. Figure 8
presents $x-r_{c}$ phase plane under different $\delta$ and changing speed of
environment $a$. Results show that all subfigures have a large range of orange
trajectories eventually evolving along an interior closed orbit, although the
shapes of orange range are distinguished in different figures. This indicates
that the interior closed orbit always emerges for a large range of $\delta$
and that the value of $\delta$ can alter the attraction range of the interior
closed orbit.

Fig 8: Game-environment evolution under different initial phase $\bm{\delta}$
and self-changing speed $\bm{a}$ of continuously changing global environment.
In $x-r_{c}$ plane, there are always three kinds of trajectories: the blue,
pink and orange trajectories will eventually evolve to $\left(x^{*}=0\right)$,
$\left(x^{*}=1,r_{c}^{*}=\alpha\right)$ and circulate along a closed orbit,
respectively. In all panels, $N=4$, $\alpha=1.5$, $\beta=3.5$, $\theta=0.5$,
$r_{d}=0.6$, $\epsilon=7$.

## Appendix C. Discretely changing global environment: four-segment function

In this part, we consider a four-segment function $w_{3}(t)$ to describe the
discretely changing global environments, which aims to check the explanations
about why there emerges interior closed orbit. Specifically, the formula of
$w_{3}(t)$ is

$\displaystyle{w_{\rm{3}}}\left(t\right)=\left\\{{\begin{array}[]{*{20}{c}}{1.4}&{\left[t/T\right]=4n}\\\
{1}&{\left[t/T\right]=4n+1}\\\ {0.6}&{\left[t/T\right]=4n+2}\\\
{1}&{\left[t/T\right]=4n+3}\\\ \end{array}}\right.,n=0,1,2,\cdots\cdots$ (46)

Figure 9 presents game-environment evolution under discretely changing
environment given by $w_{3}\left(t\right)$. Results also show the emergence of
cyclic evolutions (represented by orange trajectories in Figure 9(a)). We
present some representative trajectories through selecting different initial
points in Figure 9(b). Noteworthy, the orange and pink trajectories display
the status of cyclic evolution of group cooperation and local environment.
Take the orange trajectory as an example. The trajectory approaches fixed
point 1 (labelled in figure 9(b)) during the first $T$ time, then seized by
fixed point 2 when the global environment changes to the second stage. When
the global environment evolves to the third stage, unexpectedly, the dynamical
trajectory is not actually captured but is still attracted by fixed point 3,
which subsequently be seized by fixed point 2 again. Finally, as fixed point 2
is within the attract domain of fixed point 1, the trajectory forms a complete
loop, leading to the cyclic evolution along an interior closed orbit
consisting of four path segments. The time-evolution pictures (see Figure
9(c)(d)) clearly present the periodic behaviors of $x$ and $r_{c}$. This
complicated evolutionary process, which shows the important characteristics
emerged both in discrete environment $w_{1}(t)$ where fixed point 1 and fixed
point 2 are in the mutual attraction domains and in continuous environment
$w_{2}(t)$ where the dynamical path are not captured yet are strongly
attracted by the fixed point, supports our limit thought which explains the
formation of the closed orbit under continuously changing environment to some
extent.

Fig 9: Emergence of cyclic evolutions of group cooperation and local
environment under periodically changing global environments given by
$\bm{w_{3}\left(t\right)}$. (a) Local game-environment evolutions. (b) Four
typical dynamic trajectories with different initial conditions. (c-d) Time
evolutions of $x$ and $r_{c}$, corresponding to the colored trajectories in
(b). Parameters: $N=4$,$\alpha=1.5$, $\beta=3.5$, $\theta=1$, $r_{d}=1.2$,
$a=0.1$, $\epsilon=4$.