# The paradigm of tax-reward and tax-punishment strategies in the advancement
of public resource management dynamics

Lichen Wang  College of Science, Northwest A & F University, Yangling 712100,
China  Yuyuan Liu  College of Science, Northwest A & F University, Yangling
712100, China  Ruqiang Guo  College of Science, Northwest A & F University,
Yangling 712100, China  Liang Zhang  College of Science, Northwest A & F
University, Yangling 712100, China  Linjie Liu<EMAIL_ADDRESS>College of
Science, Northwest A & F University, Yangling 712100, China  College of
Economics & Management, Northwest A&F University, Yangling 712100, China
Shijia Hua<EMAIL_ADDRESS>College of Science, Northwest A & F University,
Yangling 712100, China

###### Abstract

In contemporary society, the effective utilization of public resources remains
a subject of significant concern. A common issue arises from defectors seeking
to obtain an excessive share of these resources for personal gain, potentially
leading to resource depletion. To mitigate this tragedy and ensure sustainable
development of resources, implementing mechanisms to either reward those who
adhere to distribution rules or penalize those who do not, appears
advantageous. We introduce two models: a tax-reward model and a tax-punishment
model, to address this issue. Our analysis reveals that in the tax-reward
model, the evolutionary trajectory of the system is influenced not only by the
tax revenue collected but also by the natural growth rate of the resources.
Conversely, the tax-punishment model exhibits distinct characteristics when
compared to the tax-reward model, notably the potential for bistability. In
such scenarios, the selection of initial conditions is critical, as it can
determine the system’s path. Furthermore, our study identifies instances where
the system lacks stable points, exemplified by a limit cycle phenomenon,
underscoring the complexity and dynamism inherent in managing public resources
using these models.

The rampant overexploitation of public resource pools has emerged as a
pervasive issue in today’s world, cutting across geographical and cultural
boundaries plus1 ; plus2 ; plus3 ; gg1 . As humanity’s footprint on ecosystems
escalates, the reciprocal influence of ecological variables on human societies
is concurrently intensifying. This intricate interplay between societal and
ecological systems is not an isolated incident but a globally prevalent
phenomenon liu2023coevolutionary ; rl2 ; third2 ; rll1 . A case in point is
the widespread overgrazing of public rangelands. Across numerous regions,
rangeland resources are subjected to relentless exploitation, culminating in
their alarming depletion gdfm1 ; gdfm2 . This predicament frequently arises in
the absence of effective management and vigilant supervision, as herders,
driven by their necessities, overgraze the land, thereby impeding the
resources’ ability to rejuvenate, and propelling them towards eventual
exhaustion. This scenario is not confined to rangelands alone but is mirrored
in a multitude of contexts, such as the overfishing of marine resources, the
excessive consumption of water resources, the exacerbation of global climate
change ref21 ; ref19 ; ref16 ; ref17 ; wang2020communicating . These instances
underscore the urgent need for comprehensive strategies to manage public
resources sustainably.

To address the challenge of resource overexploitation, managers often
establish allocation rules. Despite these efforts, individual desires for
greater benefits frequently lead to the violation of these rules, exacerbating
resource depletion. This scenario underscores the significance of cooperative
behavior, wherein participants comply with allocation rules out of a
collective commitment to resource sustainability. This raises a pivotal
question: How can we motivate individuals to persist in such cooperative
behavior for the enduring sustainability of resources? This inquiry has
generated substantial interest and debate in the field hz1 ; hz2 ; hz3 ;
capraro2021mathematical .

The intricacies of public resource management often call for the adoption of
reward and punishment strategies, a stance substantiated by a wealth of
studies sigmund2010social ; hilbe2010incentives ; ref4 ; ref5 ; ref7 ; ref8 ;
ref10 ; ref12 ; ref13 ; jl1 ; wang2023emergence ; jl2 . In particular, the
work of Chen and Szolnoki highlights the effectiveness of punitive mechanisms
chenpcb2018 . Their inspection-punishment model, which targets and penalizes
defectors, demonstrates how different natural growth rates impact the
evolution of public resource pools. Yet, this model applies a constant
severity of punishment, while in reality, levels of punishment often adjust in
response to varying violation rates. The dynamic incentive allocation
mechanism can address these shortcomings. Under this scheme, all individuals
pay taxes. In the reward scenario, the total tax collected is evenly
distributed among all cooperators as a reward ref23 ; hua2024coevolutionary .
In the punishment scenario, the total tax is evenly allocated to the defectors
as a penalty ref24 ; sun2021combination . However, the use of this dynamic
incentive allocation mechanism is seldom mentioned or employed in research on
public pool management. Therefore, its effectiveness in promoting the
emergence of cooperation and ensuring sustainable resource use remains to be
seen.

This study pioneers the application of a tax incentive strategy to the
management of public resource pools, utilizing replicator equations to portray
the evolutionary dynamics of cooperation ref25 ; ref26 ; ref27 ; ref28 ; ref35
. Concurrently, a modified logistic model is employed to depict the
evolutionary dynamics of public resource pools ref35 ; ref36 . We construct
tax-reward and tax-punishment models to offer a more comprehensive
understanding and prediction of the temporal evolution of public resource
pools. We take into account the finite and renewable nature of the resource
pool, along with the reciprocal influence of individual behavior on resources.
For the tax-reward strategy, we demonstrate the temporal evolution of the
proportion of cooperators and track the changes in the common resource pool,
capturing the dynamics associated with varying tax revenues and natural growth
rates. Our findings reveal that the system may ultimately gravitate towards
one of three states: full cooperation, full defection, or a coexistence of
cooperation and defection. For the tax-punishment strategy, we similarly
demonstrate the temporal evolution of the proportion of cooperators and track
the changes in the common resource pool, capturing the dynamics associated
with varying tax revenues and natural growth rates. Contrary to the tax-reward
strategy, we observe that the system may exhibit bistability, suggesting that
the stability of the system is contingent upon initial conditions.
Furthermore, we identify instances where stable fixed points do not exist,
such as the limit cycle discussed in this paper, representing a unique case of
unstable fixed points. Lastly, we corroborate the theoretical accuracy and the
efficacy of our model through numerical examples and Monte Carlo simulations.

Model and Methods

Let’s envision a group of $N$ individuals, all utilizing a shared, finite, and
renewable resource pool, with a natural growth rate denoted as $r$. The
current resource level of the common pool is represented by $y$. Typically,
the logistic model is used to describe the dynamic changes chenpcb2018 . The
equation for the current resource total $y$ with respect to time $t$ is as
follows:

$\displaystyle\dot{y}=ry\left(1-\frac{y}{R_{m}}\right),$ (1)

where $\dot{y}=\dfrac{dy}{dt}$, and $R_{m}$ represents the maximum capacity of
the resource environment. To ensure the sustainable development of the common
resource, a distribution rule is proposed in reference chenpcb2018 . Each
individual obtains a resource quantity of $b_{l}=\frac{b_{m}y}{R_{m}}$ from
the current resource pool, where $b_{m}$ specifies the maximum resource amount
an individual is allowed to use when the resource reaches its maximum
capacity.

To streamline our model, we posit that individuals adopt one of two possible
strategies. The first strategy, undertaken by law-abiders or cooperators,
involves adhering to established allocation rules. Conversely, the second
strategy is employed by law-breakers or defectors, who disregard these rules
in favor of personal gain, securing a portion, $b_{v}$, that surpasses the
designated quota, $b_{l}$. We define $b_{v}$ as $b_{l}(1+\alpha)$, with
$\alpha$ signifying the “defection rate”, akin to the dilemma strength often
mentioned in prior research on $2\times 2$ and $N$-player games
arefin2020social ; wang2015universal ; ito2018scaling ;
tanimoto2007relationship . In this scenario, we presume that an individual’s
payoff is equivalent to the share of resources they extract from the
collective resource pool.

In analyzing a well-mixed population, we employ replicator equations to model
the temporal evolution of competing strategies. The fraction of cooperators
within the population is denoted by $x$; consequently, we can assert the
governing equation as follows:

$\displaystyle\dot{x}=x(1-x)\left(P_{C}-P_{D}\right),$ (2)

wherein $\dot{x}$ signifies the rate of change of $x$ concerning time $t$,
symbols $P_{C}$ and $P_{D}$ pertain to the average payoffs for cooperators and
defectors, correspondingly.

As the concept of coevolution accounts for the reciprocal influence of
individual behavior on the prevailing resource conditions, the governing
equation governing the abundance of shared resources can be extended to:

$\displaystyle\dot{y}=ry\left(1-\frac{y}{R_{m}}\right)-N\left[b_{l}x+(1-x)b_{v}\right].$
(3)

To avoid cooperators turning into defectors due to unfair distribution of
benefits, leading to the depletion of common resources, we propose the
following two incentive strategies to promote the sustainable development of
common resources.

Tax-reward

In the context of real-world resource management, rewarding cooperators is
generally considered an effective strategy. Therefore, we propose a tax-reward
strategy. In this strategy, we assume a population consisting of $N$
individuals, with $N_{C}$ being the number of cooperators. Each individual is
required to pay a tax of $\delta$. The total revenue generated from these
taxes is then used to provide rewards to the cooperators. The average payoff
for cooperators, denoted as $P_{C}$, and the average payoff for defectors,
denoted as $P_{D}$, can be expressed as follows:

$\displaystyle\begin{split}P_{C}&=\sum_{k=0}^{N-1}\binom{N-1}{k}x^{k}(1-x)^{N-1-k}\left(b_{l}-\delta+\frac{N\delta}{k+1}\right),\\\
P_{D}&=\sum_{k=0}^{N-1}\binom{N-1}{k}x^{k}(1-x)^{N-1-k}\left[b_{l}(1+\alpha)-\delta\right].\end{split}$
(4)

According to the property of binomial coefficients
$\binom{N-1}{k}\frac{N}{k+1}=\binom{N}{k+1}$, we can obtain:

$\displaystyle\begin{split}P_{C}-P_{D}&=\sum_{k=0}^{N-1}\binom{N-1}{k}x^{k}(1-x)^{N-1-k}\left(-b_{l}\alpha+\frac{N\delta}{k+1}\right)\\\
&=-\frac{b_{m}y}{R_{m}}\alpha+\delta\sum_{k=0}^{N-1}\binom{N}{k+1}x^{k}(1-x)^{N-1-k}\\\
&=-\frac{b_{m}y}{R_{m}}\alpha+\delta\sum_{k^{\prime}=1}^{N-1}\binom{N}{k^{\prime}}x^{k^{\prime}-1}(1-x)^{N-k^{\prime}}\\\
&=-\frac{b_{m}y}{R_{m}}\alpha+\delta\sum_{k^{\prime}=1}^{N-1}\binom{N}{k^{\prime}}\frac{x^{k^{\prime}}(1-x)^{N-k^{\prime}}}{x}\\\
&=-\frac{b_{m}y}{R_{m}}\alpha+\delta\frac{1-(1-x)^{N}}{x}.\end{split}$ (5)

Based on recent dynamic feedback modeling approaches chenpcb2018 ;
arefin2021imitation , we derived our tax-reward model as follows:

$\displaystyle\left\\{\begin{array}[]{l}\dot{x}=x(1-x)\left[\delta\frac{1-\left(1-x\right)^{N}}{x}-\frac{y}{R_{m}}b_{m}\alpha\right],\\\
\dot{y}=ry\left(1-\frac{y}{R_{m}}\right)-N\frac{y}{R_{m}}b_{m}[1+(1-x)\alpha].\end{array}\right.$
(8)

Tax-punishment

In real-life situations, when the proportion of cooperators is relatively
high, the cost of rewarding them can often be substantial. In such cases, it
becomes necessary to consider the punishment of defectors. We propose a tax-
punishment strategy. Firstly, let $N_{D}$ denote the number of defectors among
the $N$ individuals in the population. Each individual is required to pay a
tax of $\delta$. The total revenue generated from these taxes is then utilized
to administer punishment to the defectors. Similarly, the average payoff for
cooperators, denoted as $P_{C}$, and the average payoff for defectors, denoted
as $P_{D}$, can be expressed as follows:

$\displaystyle\begin{split}P_{C}&=\sum_{k=0}^{N-1}\binom{N-1}{k}x^{k}(1-x)^{N-1-k}\left(b_{l}-\delta\right),\\\
P_{D}&=\sum_{k=0}^{N-1}\binom{N-1}{k}x^{k}(1-x)^{N-1-k}\left[b_{l}(1+\alpha)-\delta-\frac{N\delta}{N_{D}+1}\right].\end{split}$

According to the property of binomial coefficients
$\binom{N-1}{k}\frac{N}{N-k}=\binom{N}{k}$, we can obtain:

$\displaystyle\begin{split}P_{C}-P_{D}&=\sum_{k=0}^{N-1}\binom{N-1}{k}x^{k}(1-x)^{N-1-k}\left(-b_{l}\alpha+\frac{N\delta}{N-k}\right)\\\
&=-\frac{b_{m}y}{R_{m}}\alpha+\delta\sum_{k=0}^{N-1}\binom{N}{k}x^{k}(1-x)^{N-1-k}\\\
&=-\frac{b_{m}y}{R_{m}}\alpha+\delta\sum_{k=0}^{N-1}\binom{N}{k}\frac{x^{k}(1-x)^{N-k}}{1-x}\\\
&=-\frac{b_{m}y}{R_{m}}\alpha+\delta\frac{1-x^{N}}{1-x}.\end{split}$

Based on the above, we obtained our tax-punishment model as follows:

$\displaystyle\left\\{\begin{array}[]{l}\dot{x}=x(1-x)\left(\delta\frac{1-x^{N}}{1-x}-\frac{y}{R_{m}}b_{m}\alpha\right),\\\
\dot{y}=ry\left(1-\frac{y}{R_{m}}\right)-N\frac{y}{R_{m}}b_{m}[1+(1-x)\alpha].\end{array}\right.$
(11)

To help readers easily understand all the parameters and variables introduced
in our work, we present them in Table 1.

Table 1: Symbols and meanings used in this article. Symbol |  Meaning
---|---
$N$ |  Number of individuals in the game group
$N_{C}$ |  Number of cooperators
$N_{D}$ |  Number of defectors
$y$ |  Current environmental resource level
$x$ |  Current proportion of cooperators
$\alpha$ |  Defection rate of defectors
$\delta$ |  Tax paid by each participant
$R_{m}$ |  Maximum resource carrying capacity of the environment
$b_{m}$ |  Maximum amount of resources allowed for an individual when resources reach maximum capacity
$b_{l}$ |  Amount of resources obtained by cooperators
$b_{v}$ |  Amount of resources obtained by defectors
$P_{C}$ |  Average payoff of cooperators
$P_{D}$ |  Average payoff of defectors

Results

Tax-reward

For the tax-reward model, utilizing equation (8), there are four boundary
fixed points in the system: $\left(x,y\right)=\left(0,0\right)$,
$\left(1,0\right)$, $\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$, and
$\left(0,R_{m}-\frac{Nb_{m}}{r}\left(1+\alpha\right)\right)$. Additionally,
when the conditions $\delta-\alpha b_{m}+\frac{\alpha Nb_{m}^{2}}{rR_{m}}<0$
and $N\delta-\alpha b_{m}+\frac{\alpha
Nb_{m}^{2}}{rR_{m}}+\frac{\alpha^{2}Nb_{m}^{2}}{rR_{m}}>0$ are satisfied,
there exists one and only one interior fixed point
$\left(x^{\ast},y^{\ast}\right)$, where $0<x^{\ast}<1$ and $0<y^{\ast}<R_{m}$.
otherwise, the system does not have an interior fixed point. We use the sign
of the eigenvalues of the Jacobian matrix to determine the stability of the
fixed points (detailed proof can be found in Supplementary Information). Now
we present the evolution of cooperation proportion and resource abundance over
time for different initial conditions, along with their corresponding phase
diagrams where the solid black dots represent stable fixed points , while the
hollow black dots represent unstable fixed points. Here, we define
$E_{C}=\frac{b_{m}N}{Rm}$ and $E_{D}=\frac{b_{m}N(1+\alpha)}{R_{m}}$. We will
discuss the following three cases based on different natural growth rates $r$.

Slowly Growing Resource Pool

In the case of slow resource growth, when $0<r<E_{C}<E_{D}$. Since $x\in[0,1]$
and $y\in[0,R_{m}]$, the system has exactly two boundary fixed points, $(0,0)$
and $(1,0)$. It is evident that, by analyzing the sign of the eigenvalues,
$(0,0)$ is unstable, while $(1,0)$ is stable. Figure 1 elegantly illustrates
the dynamic interplay between the cooperation rate and the abundance of the
shared resource pool over time, under the conditions where $0<r<E_{C}<E_{D}$.
This captivating depiction is further complemented by an accompanying phase
diagram, which effectively encapsulates the interaction of these variables
within the specified parameter space. From the figure, an important result can
be observed: when the natural growth rate $r$ is low, the reward mechanism
drives participants towards full cooperation. However, even with significant
rewards for cooperation, the system’s resources will eventually be completely
depleted, indicating that the resources are not sustainable. In this case,
cooperative behavior may lose its value due to resource exhaustion. Therefore,
it is advisable to avoid exploiting this type of common resource pool.

Moderately growing resource pool

In the case of moderate resource growth, when $0<E_{C}<r<E_{D}$. We have
$N\delta-\alpha b_{m}+\frac{\alpha
Nb_{m}^{2}}{rR_{m}}+\frac{\alpha^{2}Nb_{m}^{2}}{rR_{m}}>0$. We find that
depending on the value of tax revenue $\delta$, the system may stabilize at
different fixed points. When $\delta-\alpha b_{m}+\frac{\alpha
Nb_{m}^{2}}{rR_{m}}>0$, the system has only three boundary fixed points,
$\left(0,0\right)$, $\left(1,0\right)$, and
$\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$. By analyzing the signs of the
eigenvalues, we determine that in this case, the system has a unique stable
fixed point at $\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$. It can be observed
that when tax revenue $\delta$ is high, the system eventually leads to all
participants transforming into cooperators who share the common resource pool,
allowing for sustainable development of the resource pool. This intriguing
scenario unfolds vividly in the premier column of Figure 2.

In the event that the expression $\delta-\alpha b_{m}+\frac{\alpha
Nb_{m}^{2}}{rR_{m}}$ is less than zero, the system elegantly unfolds to reveal
three boundary fixed points of notable significance: $\left(0,0\right)$,
$\left(1,0\right)$, and $\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$, as well as
one interior fixed point $\left(x^{\ast},y^{\ast}\right)$. According to the
sign analysis of eigenvalues, the first three boundary fixed points are
unstable, while the last interior fixed point is stable (the stability proof
for interior points can be found in Supplementary Information). It can be
observed that when the tax revenue $\delta$ is low, the system will eventually
reach a state of coexistence between cooperators and defectors. In this case,
the resource environment can also be sustainably utilized. This situation is
shown in the second column of Figure 2.

Rapidly growing resource pool

In the case of rapid resource growth, when $r>E_{D}$. We find that the choice
of tax revenue $\delta$ is crucial. Depending on the different levels of tax
revenue, the system may stabilize at different fixed points. However, due to
the high natural growth rate of resources, the resource level in the system
will always be maintained at a relatively high level for sustainable
development. The stability results of the system corresponding to different
levels of tax revenue are provided below.

When $N\delta-\alpha b_{m}+\frac{\alpha
Nb_{m}^{2}}{rR_{m}}+\frac{\alpha^{2}Nb_{m}^{2}}{rR_{m}}<0$, the system
elegantly unveils itself to possess merely four boundary fixed points. These
points, each a testament to the system’s inherent intricacy, are as follows:
$\left(x,y\right)=\left(0,0\right)$, $\left(1,0\right)$,
$\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$, and
$\left(0,R_{m}-\frac{Nb_{m}}{r}\left(1+\alpha\right)\right)$. Through our
theoretical analysis, we ascertain that the fixed point
$\left(0,R_{m}-\frac{Nb_{m}}{r}\left(1+\alpha\right)\right)$ is stable, while
the other fixed points are unstable. This is an interesting result, as with
low tax revenue $\delta$, all participants in the system eventually become
defectors. However, due to the high natural growth rate of resources, the
resource level in the system is maintained at a relatively high level for
sustainable development. This situation is shown in the first column of Figure
3.

When $\delta-\alpha b_{m}+\frac{\alpha Nb_{m}^{2}}{rR_{m}}>0$, the system, in
all its complexity, elegantly presents a quartet of boundary fixed points.
These points, each a testament to the system’s elaborate dynamics, are
distinctly located at: $\left(x,y\right)=\left(0,0\right)$,
$\left(1,0\right)$, $\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$,
$\left(0,R_{m}-\frac{Nb_{m}}{r}\left(1+\alpha\right)\right)$. It can be proven
that the fixed point $\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$ is stable, while
the other fixed points are unstable. In this case, we can observe that when
the tax revenue $\delta$ is relatively high, all participants in the system
will eventually become cooperators, and the resource level in the system will
be maintained at a relatively high level for sustainable development. This
situation is shown in the second column of Figure 3.

When $\delta-\alpha b_{m}+\frac{\alpha Nb_{m}^{2}}{rR_{m}}<0<N\delta-\alpha
b_{m}+\frac{\alpha Nb_{m}^{2}}{rR_{m}}+\frac{\alpha^{2}Nb_{m}^{2}}{rR_{m}}$,
the system has an interior fixed point $\left(x^{\ast},y^{\ast}\right)$.
Consequently, the system elegantly unfurls to reveal four boundary fixed
points: $\left(x,y\right)=\left(0,0\right)$, $\left(1,0\right)$,
$\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$,
$\left(0,R_{m}-\frac{Nb_{m}}{r}\left(1+\alpha\right)\right)$, and one interior
fixed point $\left(x^{\ast},y^{\ast}\right)$. Through analysis, we are able to
ascertain that the interior fixed point $\left(x^{\ast},y^{\ast}\right)$ is
stable (for the stability of interior fixed points, please refer to
Supplementary Information), while the four boundary fixed points are unstable.
In this case, we can observe that the system will eventually reach a state of
coexistence between cooperators and defectors. In this state, the resource
level in the system will be maintained at a relatively high level for
sustainable development. This situation is shown in the third column of Figure
3.

Tax-punishment

Drawing parallels to the scenario involving rewards, an examination of the
tax-punishment strategy through the lens of equation (11) unveils a similar
pattern. The system, in its intricate dance of numbers, reveals four boundary
fixed points: $\left(x,y\right)=\left(0,0\right)$, $\left(1,0\right)$,
$\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$,
$\left(0,R_{m}-\frac{Nb_{m}}{r}\left(1+\alpha\right)\right)$. The detailed
existence conditions and stability analysis of the interior equilibrium points
are presented in the Supplementary Information. Based on the different natural
growth rate $r$, we will discuss the following three cases.

Slowly Growing Resource Pool

In the case of slow resource growth, that is $0<r<E_{C}<E_{D}$, the system has
two and only two equilibrium points: $(0,0)$ and $(1,0)$. Obviously, by
analyzing the signs of the eigenvalues, we can determine that $(0,0)$ is
unstable, while $(1,0)$ is stable. Figure 4 elegantly unfurls the tapestry of
time, charting the evolutionary trajectory of the proportion of cooperators
and the abundance of the common resource pool. This intricate dance of
variables plays out under the condition where $0<r<E_{C}<E_{D}$. The figure
also gracefully presents the corresponding phase diagram, a testament to the
system’s dynamism under the specified parameters. This visual representation
serves as a window into the system’s complex interplay of factors,
illuminating the intricate relationships that govern its behavior. Similar to
the case of tex-reward, when the natural growth rate $r$ is low, the punishing
mechanism drives the participants towards complete cooperation. However, even
with significant penalties imposed on defectors, the resources will eventually
be completely depleted. In this scenario, high levels of cooperation become
meaningless.

Moderately growing resource pool

When resource growth is moderate, specifically $0<E_{C}<r<E_{D}$. For the
boundary fixed point, it is evident that
$\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$ is stable when $-N\delta+\alpha
b_{m}-\frac{\alpha Nb_{m}^{2}}{rR_{m}}<0$ (More detailed theoretical analysis
can be found in the Supplementary Information). Now we provide a numerical
example. We find that the system has three boundary fixed points
$\left(0,0\right)$, $\left(1,0\right)$,
$\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$ and two interior fixed points
$\left(x_{1}^{\ast},y_{1}^{\ast}\right)$,
$\left(x_{2}^{\ast},y_{2}^{\ast}\right)$. Let’s assume
$x_{1}^{\ast}<x_{2}^{\ast}$. In this situation, the system has two stable
fixed points $\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$ and
$\left(x_{1}^{\ast},y_{1}^{\ast}\right)$, whereas the remaining fixed points
are all unstable. This scenario is illustrated in the first row of Figure 5.
We find that there are five equilibrium points in the phase plane, among which
there are two stable equilibria. One is located inside the phase plane,
indicating that cooperation can be maintained at a relatively high level and
the resource level can also remain constant. The other is located on the right
boundary of the phase plane, implying that all individuals in the group choose
cooperative behavior while resources can be sustained. It should be noted that
the emergence of one of the evolutionary outcomes depends on the initial
levels of cooperation and resources.

Rapidly growing resource pool

When the natural growth rate is high, specifically $r>E_{D}$. Clearly, when
$-N\delta+\alpha b_{m}-\frac{\alpha Nb_{m}^{2}}{rR_{m}}<0$, the fixed point
$\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$ is stable; when $\delta-\alpha
b_{m}+\frac{\alpha Nb_{m}^{2}}{rR_{m}}+\frac{\alpha^{2}Nb_{m}^{2}}{rR_{m}}<0$,
the fixed point $\left(0,R_{m}-\frac{Nb_{m}\left(1+\alpha\right)}{r}\right)$
is stable (Detailed theoretical analysis is presented in the Supplementary
Information). Here, we provide a numerical example that ensures the existence
of an unstable interior equilibrium point in the coupled system. As depicted
in the second row of Figure 5, there exist five equilibrium points in the
phase plane: $\left(0,0\right)$, $\left(1,0\right)$,
$\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$,
$\left(0,R_{m}-\frac{Nb_{m}\left(1+\alpha\right)}{r}\right)$, and one interior
fixed point $\left(x^{\ast},y^{\ast}\right)$. In this situation, the two
boundary fixed points $\left(1,R_{m}-\frac{Nb_{m}}{r}\right)$ and
$\left(0,R_{m}-\frac{Nb_{m}\left(1+\alpha\right)}{r}\right)$ are stable, but
the interior fixed point $\left(x^{\ast},y^{\ast}\right)$ is unstable. Similar
to the scenario with a moderate natural growth rate, the system displays
bistable outcomes, underscoring the importance of the selection of initial
conditions, as the system’s stability hinges on these chosen values. Despite
the potential for the system to ultimately evolve towards either complete
cooperation or complete defection, the high natural growth rate of the common
resource pool enables the system to consistently maintain a non-zero level of
resources.

It is worth noting that the coupled system can generate limit cycle dynamics.
Figure 6 shows the representative evolution of cooperation level and abundance
of the common resource pool over time, as well as the phase diagram of
$x-y/Rm$ under the condition of $E_{C}<r<E_{D}$ (The theoretical analysis can
be found in Case 1 of the Supplementary Information). We observe four
equilibrium points in the phase plane, of which three are boundary equilibrium
points and one is an interior equilibrium point, all of which are unstable.
Trajectories originating from any initial point in the phase plane ultimately
converge to a limit cycle within the plane (see Figure 6B). This implies that
the frequency of cooperators and the level of resources exhibit oscillating
evolutionary dynamics. The trajectory of system states over time in Figure 6A
also verifies this result.

Monte Carlo Simulations

In our quest to corroborate the efficacy of our model, we embarked on a series
of Monte Carlo numerical simulations. The application of Monte Carlo
methodologies within the framework of system dynamics serves as a powerful
tool to augment our comprehension of system behavior, and more importantly, to
bolster our predictive prowess. This, in turn, equips decision-makers with the
capacity to devise more potent strategies and to appraise the risks and
uncertainties intertwined with their decisions with greater precision. In the
forthcoming section, we will elucidate the robustness and effectiveness of our
dual-model framework, as substantiated by our comprehensive Monte Carlo
simulations.

Tax-reward

For each individual $i$ in the population, they are required to pay taxes
$\delta$, which will be evenly distributed as reward to cooperators. Each
individual has only two choices: to become a cooperator (denoted as $s_{i}=1$)
or become a defector (denoted as $s_{i}=0$). If individual $i$ is a
cooperator, his/her payoff is
$F_{i}=b_{m}y(t)/R_{m}-\delta+N\delta/\sum_{i=1}^{N}s_{i}$. Otherwise, if
individual $i$ is a defector, his/her payoff is
$F_{i}=\left(1+\alpha\right)b_{m}y(t)/R_{m}-\delta$. The update of the common
resource level $y(t)$ is governed by the following rule:

$\displaystyle
y(t+1)=y(t)+ry(t)\left[1-\frac{y(t)}{R_{m}}\right]-\sum_{i=1}^{N}\left[s_{i}\frac{y(t)b_{m}}{R_{m}}+\left(1-s_{i}\right)\frac{y(t)b_{m}(1+\alpha)}{R_{m}}\right].$
(12)

The selection of individual strategies involves each individual $i$ learning
from a random individual $j$. If the payoff $F_{i}$ for individual $i$ under
his/her chosen strategy is lower than the payoff $F_{j}$ for individual $j$,
then individual $i$ will adopt the strategy of $j$ with a probability $q$.
Otherwise, individual $i$ will remain with their current strategy. We refer to
the probability $q$ as the transition probability.

$\displaystyle q=\frac{F_{i}-F_{j}}{M},$ (13)

where $M$ quantifies the uncertainty in strategy adoption.

Figure 7 illustrates the results of the evolution of a resource pool over time
under different scenarios: slow growth, moderate growth, and rapid growth, as
obtained from Monte Carlo simulations. From the figures, it can be observed
that the system eventually stabilizes at the theoretically calculated fixed
points. In the case of slow resource growth, the magnitude of rewards is
insufficient to sustain resource development. Although all participants
eventually transition to cooperators, the resources will be completely
depleted (see Figure 7 A). Under moderate resource growth, consistent results
with the theoretical expectations are obtained, where the stability of the
system is determined by the magnitude of rewards (see Figure 7 B and C). In
the scenario of rapid resource growth, depending on the reward levels, the
system may reach a state of complete defection (see Figure 7 D), complete
cooperation (see Figure 7 E), or coexistence of cooperation and defection (see
Figure 7 F). Irrespective of which among the aforementioned three scenarios
manifests, the resources in the public pool consistently maintain a stable
level.

Tax-punishment

For the tax-punishment strategy, if individual $i$ is a cooperator, his/her
payoff is $F_{i}=b_{m}y(t)/R_{m}-\delta$. Otherwise, if individual $i$ is a
defector, his/her payoff is
$F_{i}=\left(1+\alpha\right)b_{m}y(t)/R_{m}-\delta-N\delta/\sum_{i=1}^{N}(1-s_{i})$.
We still adopt the same updating strategy for the common resource pool and
individual strategies as in the case of reward.

In Figure 8, we present the agent-based simulation results considering
scenarios with low, moderate, and high growth rates of the public resource
pool, respectively, when taking into account tax-punishment. Our simulation
results corroborate the theoretical findings: when the growth rate of the
public pool is low, despite all individuals in the group ultimately opting for
a cooperative strategy, the resources inevitably reach a state of complete
depletion (see Figure 8A). This implies that even universal cooperation cannot
overcome the tragedy of the commons induced by an excessively low resource
growth rate. When the growth rate of the public pool is moderate, depending on
the initial conditions, the system either stabilizes in a state where
cooperators and defectors coexist sustainably with resources, or it stabilizes
in a state where all individuals choose cooperation accompanied by sustainable
resources (see Figure 8B). Either outcome results in a situation where both
cooperation and resources are maintained. When the growth rate of the public
pool is high, another bistable outcome can emerge: regardless of whether the
population ultimately evolves into a state of complete cooperation or complete
defection, resources can be sustained (see Figure 8C). Due to the high natural
growth rate of resources in this case, the resource level of the system can be
sustained at a higher level and be developed sustainably.

Conclusions

The pivotal importance of sustainable resource development is gaining
heightened acknowledgment in our contemporary society. Consequently,
forecasting the enduring impacts of individual actions on resource stewardship
has ascended to the forefront as a paramount challenge. Existing research has
explored the complex relationship between resource environments and
stakeholder behaviors, leading to the formulation of sophisticated models
traulsen2023future ; yc1 ; first2 ; second1 ; second2 ; third1 ; wang2020eco .
These models integrate diverse factors such as incentives chenpcb2018 ,
regulatory norms ostrom1990governing , and sociocultural influences
leach1999environmental , providing a comprehensive perspective on resource
utilization dynamics. Although relevant research has been initiated, the
understanding of how to better design incentives to regulate individual
behaviors for promoting resource sustainability remains in a nascent stage.

We have proposed new incentive strategies and established tax-reward and tax-
punishment models to more accurately describe the reality of resource
development. We have found that the effects generated by rewarding cooperators
and punishing defectors are not uniform. Specifically, in scenarios
characterized by a low natural growth rate of resources, neither the tax-
reward nor the tax-punishment model is capable of achieving sustainable
resource development solely through rewarding cooperators or punishing
defectors. Regardless of the magnitude of rewards given to cooperators or
punishments imposed on defectors, although participants in the system tend to
become all cooperators, the resources in the environment will eventually
deplete. Therefore, for resources with low natural growth rates, more
effective protection measures should be implemented, such as avoiding
exploitation, to prevent eventual resource depletion. For resources with high
natural growth rates, our research results in the tax-reward model have
demonstrated that sustainable resource development can be ensured. Here the
magnitude of rewards does not affect the sustainability of resources but only
the total resource level in the steady state. This provides important insights
for practical applications: by adjusting the size of the tax payment $\delta$,
the system can eventually stabilize at a higher resource level. Regarding the
tax-punishment model, our findings suggest the potential for bistability
within the system. The stability of the fixed points is not solely tethered to
the magnitude of the tax payment, denoted as $\delta$, but also intricately
intertwined with the choice of the initial state. This dual dependency adds a
layer of complexity to the system’s dynamics. In this case, choosing an
appropriate initial state is also critical in maintaining a high resource
level. For a resource pool with moderate natural growth rates, our research
results in the tax-reward model show that the system only exhibits one stable
fixed point. In reality, we need to adjust the tax revenue $\delta$ to
maintain a higher resource level in the system. In our exploration of the tax-
punishment model, we have discovered that the system may display bistability,
suggesting that the system’s stability is contingent upon the strength of
punishment and the selection of the initial states. However, it’s important to
acknowledge that in the tax-punishment model, scenarios may arise where stable
fixed points are absent. For instance, this paper presents an example of a
limit cycle, wherein the system fails to achieve a stable state and instead
undergoes periodic cycling. This phenomenon could stem from intricate internal
interactions and feedback loops within the system. This insight is crucial for
gaining a more profound understanding of the system’s dynamic behavior and for
predicting future evolutionary trends.

Our research results provide important theoretical foundations for decision-
making in practical contexts, aiding us in better resource management and
utilization to achieve sustainable development. However, we also recognize
that the model has its inherent limitations. For instance, we assume that all
individuals are perfectly rational, while in reality, individual decision-
making may be influenced by various factors. Therefore, future research can
further expand and optimize our models, such as introducing more real-life
factors, considering irrationality in individual behavior hua2023evolution ;
sarkar2023managing ; hilbe2018evolution , accounting for resource growth rates
varying with the current resource level $y$, or incorporating the migration of
individuals wang2016effects ; shu2023determinants . It would also be
beneficial to compare the actual cost-effectiveness of the tax-reward and tax-
punishment mechanisms proposed in our current work. Optimal institutional
incentives for promoting cooperation, as studied in recent works within the
context of public goods and prisoner’s dilemma games, have shown that adaptive
rewarding strategies considering population statistics, such as the threshold
number of cooperators in the population, or local network properties, like the
number of cooperators in the network, could be more cost-efficient in
promoting cooperation han2022institutional ; duong2021cost ; wang2019exploring
; cimpeanu2023does ; duong2023cost . Exploring the applicability of such
adaptive approaches in our current context, depending on the resource growth
rate and resource dynamics, could provide further insights into effective
incentive mechanisms for sustainable resource management.

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Ethics.
This work did not require ethical approval from a human subject or animal
welfare committee.

Acknowledgments
This research was supported by the National Natural Science Foundation of
China (Grant No. 62306243), the Natural Science Foundation of Shaanxi (Grant
Nos. 2023-JC-QN-0791, 2023-JC-YB-083, 23JK0693, and 23JK0692) and the
Fundamental Research Funds of the Central Universities of China (Grants No.
2452022012, 2452022144).

Data accessibility
All pertinent analysis has been included in the Supplementary Information.
Source code is available at the Dryad, Dataset,
https://doi.org/10.5061/dryad.pnvx0k6w6.

Declaration of AI use
We have not used AI-assisted technologies in creating this article.

Author contributions
L. W.: investigation, methodology, writing—original draft, writing—review and
editing; Y. L.: investigation, methodology; R. G.: investigation; L. Z.:
writing—original draft; L. L: conceptualization, formal analysis,
investigation, supervision, methodology, writing—original draft,
writing—review and editing; S. H.: formal analysis, methodology, supervision,
writing—original draft, writing—review and editing.

Competing financial interests
We declare we have no competing interests.

Figure 1: The co-evolutionary dynamics of dynamic feedback games when the
growth rate of the resource pool is slow. Panel A presents the evolution of
the system’s state over time. Panel B shows the Phase diagram of $x-y/R_{m}$.
Parameters are: $r=0.25$, $\delta=0.2$, $N=1000$, $R_{m}=1000$, $\alpha=0.5$,
and $b_{m}=0.5$. Figure 2: The co-evolutionary dynamics of dynamic feedback
games when the growth rate of the resource pool is moderate. Panel A and C
presents the evolution of the system’s state over time. Panel B and D shows
the Phase diagram of $x-y/R_{m}$. The parameters for panel A and C are:
$r=0.6$, $\delta=0.2$, $N=1000$, $R_{m}=1000$, $\alpha=0.5$, and $b_{m}=0.5$.
The parameters for panel B and D are: $r=0.6$, $\delta=0.02$, $N=1000$,
$R_{m}=1000$, $\alpha=0.5$, and $b_{m}=0.5$. Figure 3: The co-evolutionary
dynamics of dynamic feedback games when the growth rate of the resource pool
is high. Panel A, B and C presents the evolution of the system’s state over
time. Panel D, E and F shows the Phase diagram of $x-y/R_{m}$. The parameters
for panel A and D are $r=1.0$, $\delta=0.00002$, $N=1000$, $R_{m}=1000$,
$\alpha=0.5$, and $b_{m}=0.5$. The parameters for panel B and E are $r=1.0$,
$\delta=0.2$, $N=1000$, $R_{m}=1000$, $\alpha=0.5$, and $b_{m}=0.5$. The
parameters for panel C and F are $r=1.0$, $\delta=0.04$, $N=1000$,
$R_{m}=1000$, $\alpha=0.5$, and $b_{m}=0.5$. Figure 4: The co-evolutionary
dynamics of coupled systems, under sscenarios of a slowly growing resource
pool, incorporate tax-punishment. Panel A presents the evolution of the
system’s states over time. Panel B shows the Phase diagram of $x-y/R_{m}$.
Parameters are $r=0.25$, $\delta=0.2$, $N=1000$, $R_{m}=1000$, $\alpha=0.5$,
and $b_{m}=0.5$. Figure 5: The coupled system can produce bistable outcomes.
Panels A, B, D and E presents the evolution of the system’s state over time.
Panel C and F shows the Phase diagram of $x-y/R_{m}$. The parameters for
panels A, B, and C are $r=0.6$, $\delta=0.004$, $N=1000$, $R_{m}=1000$,
$\alpha=0.5$, and $b_{m}=0.5$. The parameters for panels D, E and F are
$r=0.9$, $\delta=0.003$, $N=1000$, $R_{m}=1000$, $\alpha=0.5$, and
$b_{m}=0.5$. Figure 6: The coupled system can generate evolutionary
oscillatory dynamics. Panel A presents the evolution of the system’s state
over time. Panel B shows the Phase diagram of $x-y/R_{m}$. Parameters are
$r=0.006$, $\delta=0.0001$, $N=10$, $R_{m}=1000$, $\alpha=0.5$, and
$b_{m}=0.5$. Figure 7: The results of Monte Carlo simulations for the coupled
system with tax-reward under three distinct scenarios. The solid black dots in
the figures represent the calculated stable fixed points. Panel A displays the
simulation outcomes when the resource growth rate is low, while Panels B and C
present two different sets of results for scenarios with a moderate resource
growth rate. Panels D, E, and F exhibit three distinctly varied simulation
outcomes under conditions of a high resource growth rate. The parameters for
panel A are $M=1$, $r=0.25$, $\delta=0.2$, $N=1000$, $R_{m}=1000$,
$\alpha=0.5$, and $b_{m}=0.5$. The parameters for panel B are $M=1$, $r=0.6$,
$\delta=0.2$, $N=1000$, $R_{m}=1000$, $\alpha=0.5$, and $b_{m}=0.5$. The
parameters for panel C are $M=0.1$, $r=0.6$, $\delta=0.02$, $N=1000$,
$R_{m}=1000$, $\alpha=0.5$, and $b_{m}=0.5$. The parameters for panel D are
$M=1$, $r=1$, $\delta=0.00002$, $N=1000$, $R_{m}=1000$, $\alpha=0.5$, and
$b_{m}=0.5$. The parameters for panel E are $M=1$, $r=1$, $\delta=0.2$,
$N=1000$, $R_{m}=1000$, $\alpha=0.5$, and $b_{m}=0.5$. The parameters for
panel F are $M=1$, $r=1$, $\delta=0.04$, $N=1000$, $R_{m}=1000$, $\alpha=0.5$,
and $b_{m}=0.5$. Figure 8: Monte Carlo simulation results of the co-
evolutionary game model under the tax penalty scenario when considering low,
moderate, and high resource growth rates, respectively. The solid black dots
in the figures represent the calculated stable fixed points. The parameters
for panel A are $M=1$, $r=0.25$, $\delta=0.2$, $N=1000$, $R_{m}=1000$,
$\alpha=0.5$, and $b_{m}=0.5$. The parameters for panel B are $M=2$, $r=0.6$,
$\delta=0.004$, $N=1000$, $R_{m}=1000$, $\alpha=0.5$, and $b_{m}=0.5$. The
parameters for panel C are $M=1$, $r=0.9$, $\delta=0.003$, $N=1000$,
$R_{m}=1000$, $\alpha=0.5$, and $b_{m}=0.5$.