2.1 Firms

The production process involves a level of emissions that depends on the technology used by the firms. If the firm makes products using clean technology (e.g., purification systems etc.) it will incur higher production costs than if it uses less emission-conscious technology. The two different technologies entail different costs: the high-emission technology entails a production cost $c^l$ , while the legally prescribed low-emission technology entails a higher production cost $c^h$ , being $c^h\gt c^l\gt 0$ .

The firm may decide to behave dishonestly by contravening legal requirements and using the most polluting technology (prohibited by law) to incur lower production costs. The State entrusts environmental inspectors with the task of checking that firms comply with the environmental regulation.

Following previous works such as Coppier et al. (Reference Coppier, Grassetti and Michetti2021) and Panchuk et al. (Reference Panchuk, Sushko, Michetti and Coppier2022), we consider a discrete-time setup, that is, $t=0,1,2,\ldots$ , and define $x_t\in [0,1]$ as the fraction of firms making products using high-emission technology (dishonest firms) at time $t$ .

The State monitors companies to verify that they use low-emission clean technology. Let $q_t$ be the monitoring level put in place by the State at time $t$ to detect the presence of polluting firms, while $q^e_t\in [0,1]$ represents the expected probability by the firm, at any time $t$ , of being monitored, according to the expected control level fixed by the State and, then, of being reported. If a dishonest firm is monitored and detected, it is punished with a fine being a fraction $f\gt 0$ of the capital level, thus linked to the production level.

We assume for simplicity a linear technology of the kind $y_t=A k_t$ , and, without loss of generality, a unitary price of produced good. Then, by taking into account the previous considerations, we derive the expected profit at time $t$ of an honest firm (using low-emission technology) as

(1) \begin{equation} E[\pi _{h,t}] = \pi _{h,t} = Ak_t - c^hk_t, \quad A \gt c^h \gt 0. \end{equation}

On the other hand, the expected profit at time $t$ of a dishonest firm (using a high-emission technology) depends also on the event of being discovered fraudulent, that is,

(2) \begin{equation} \pi _{d,t} = \left \{ \begin{array}{l@{\quad}l} \pi _{d,NM,t}=Ak_t-c^lk_t, & \hbox{if not monitored,} \\ \pi _{d, M,t}= Ak_t-c^hk_t-fk_t, & \hbox{if monitored}, \end{array}\right . \end{equation}

being $A \gt c^l+f$ the total productivity factor. Since the monitoring level may change at any time $t$ , the expected profit at time $t$ for a dishonest firm is given by

(3) \begin{equation} E[\pi _{d,t}] = q^e_t\pi _{d, M,t}+(1-q^e_t)\pi _{d,NM,t}= Ak_t-q^e_tc^hk_t-fq^e_tk_t- (1-q^e_t)c^lk_t. \end{equation}

Therefore, the firm will find it worthwhile to pollute, that is, use high-emission technology, if

(4) \begin{equation} q^e_t\lt \frac {\Delta _c}{\Delta _c+ f}, \end{equation}

where $\Delta _c=c^h-c^l\gt 0$ .

The difference in expected profits between polluting and nonpolluting firms is then given by

(5) \begin{equation} \gamma (q^e_t) = E[\pi _{d,t}] - \pi _{h,t} = (1 - q^e_t) \Delta _c k_t - fq^e_tk_t \end{equation}

and can be rewritten in terms of a unit of capital $k_t$ , thus implying the following expression

(6) \begin{equation} \delta (q^e_t) = \frac {E[\pi _{d,t}] - \pi _{h,t}}{k_t} = (1 - q^e_t) \Delta _c - f q^e_t, \end{equation}

being a linear strictly decreasing function of the expected monitoring level. In such a way, the difference between expected profits decreases as the expected monitoring level increases, while the fine level affects the intensity of change.

2.2 Environment

We now specify the environmental quality evolution through an environmental quality index often used as a measure of the overall quality of the environment at a given time $t$ (see, e.g., John and Pecchenino (Reference John and Pecchenino1994) and Zhang (Reference Zhang1999)). This index can encompass various indicators such as air and water quality, greenhouse gas emissions, biodiversity, and ecosystem health, among others, and provides a single value that reflects the current state of environmental quality. By taking into account Grassetti et al. (Reference Grassetti, Mammana and Michetti2024), natural resources and ecosystems at time $t$ are described by the index of the environmental quality denoted by $E_t \geq 0$ .

As in Caravaggio and Sodini (Reference Caravaggio and Sodini2023), there exists a given natural level of environmental quality related to the nonpolluting case that can be asymptotically reached without human activity. Without loss of generality, we set such a level to $1$ , representing the maximum level of environmental quality that could be reached. Obviously, in the presence of polluting firms, the environmental quality will be reduced.

Similarly to what has been proposed in Caravaggio and Sodini (Reference Caravaggio and Sodini2023), by taking into account these factors, we assume that the environmental quality index evolves according to

(7) \begin{equation} E_{t+1} = E_t + b(1 - E_t) - ax_t \stackrel {df}{=} S_3(x_t, E_t), \end{equation}

where the environmental level $1$ represents the value toward that the index tends when the fraction of polluting firms is zero for all $t$ , the parameter $b \in (0, 1)$ measures the speed of reversion of the environmental quality to $1$ , and the parameter $a\in (0,b)$ measures the strength of environmental degradation due to the presence of polluting firms. Notice that condition $b \in (0, 1)$ guarantees the existence of an endogenous mechanism that leads the environment to the equilibrium level in the absence of polluting firms. In contrast, condition $a\in (0,b)$ is necessary to ensure that the environmental level cannot become negative, even in the presence of polluting firms.

2.3 State

The State monitors firms in order to detect and punish polluting behavior. Note that in previous works such as Bose et al. (Reference Bose, Capasso and Murshid2008) and Coppier et al. (Reference Coppier, Grassetti and Michetti2021) it has been assumed that the State fixes the monitoring level to be set at time $t+1$ by observing the fraction of dishonest firms at time $t$ . In a similar way, the updating control level put in place by the State is assumed to be modified (decreased or increased) starting from the existing level of audit and considering the environmental quality level in the economy to put in place an effective strategy for combating pollution.

In fact, monitoring activities require human and monetary resources that cannot be completely renewed from one period to another, there exists an “inertia” in the expenditure items of the public budget. Hence, as in Coppier et al. (Reference Coppier, Michetti and Panchuk2024), the level of control put in place is updated starting from the existing level of audit. In addition, the future level of monitoring depends on the environmental quality in the economy or the fraction of polluting firms. Since a higher fraction of polluting firms implies a lower environmental level, then we consider that the monitoring level increases as long as the environmental quality is decreased, and vice versa. As in Coppier et al. (Reference Coppier, Michetti and Panchuk2024), we assume rational expectations, that is, the State knows the future fraction of polluting firms, and hence the environmental quality level variation $E_{t+1}-E_{t}$ depending on the fraction of dishonest firms in the economy at time $t$ .

To reflect the previous arguments, a simple rule of thumb to describe the updating control strategy can be that the monitoring level is modified so that the growth rate is proportional to the change in environmental quality. To the scope we suggest the following rule

(8) \begin{equation} q_{t+1} = G(q_t, \Delta E_t) = q_t\left (1 - d\Delta E_t \right ), \end{equation}

where $\Delta E_t=E_{t+1}-E_t$ , that is, $\Delta E_t \gt 0$ if from the time $t$ to $t+1$ the environmental quality is increased, and vice versa. Parameter $d\in (0,1)$ has been introduced to capture the intensity with which $q_t$ can grow/decline from time $t$ to time $t+1$ in response to higher/lower observed environmental quality levels. In other words, the parameter $d$ measures the responsiveness of the monitoring level to changes in the environmental quality. By taking into account (7), we obtain

\begin{equation*} \Delta E_t = b(1 - E_t) - ax_t, \end{equation*}

and hence, the following rule describing the updating control by the State is reached:

(9) \begin{equation} q_{t+1} = G(x_t, q_t, E_t) = q_t\bigl [ 1 - db(1 - E_t) + da x_t \bigr ]. \end{equation}

2.4 The evolutionary model

In order to describe how the fraction of polluting firms evolves over time, we assume that they exchange information through word-of-mouth process, due to which interacting firms of different types may alter their behavior if the gains from the other approach outweighs those from their current choice.

With this aim in mind, we consider an evolutionary mechanism as firstly proposed by Banerjee and Fudenberg (Reference Banerjee and Fudenberg2004) and Dawid (Reference Dawid1999) according to which agents have the opportunity to compare their expected pay-off with those of others in society. If a firm encounters another firm exhibiting the same behavior (polluting or nonpolluting), it gains no new insights into potential payoffs and thus decides to maintain its current behavior. Conversely, a firm may opt to change its behavior (from polluting to nonpolluting, or vice versa) upon encountering a firm of a different type. After comparing their expected profits, if the firm finds that switching type could increase its own expected profits, it may choose to transition from one type to the other. Such a mechanism can be formalized as in Lamantia and Pezzino (Reference Lamantia and Pezzino2021) where it is assumed that the fraction of polluting firms at time t is updated by:

• • adding the fraction $(1-x_t)x_t\phi$ of nonpolluting firms which meet polluting ones and change type during the time interval because they learn that being polluting is more convenient;

• • subtracting the fraction $(1-x_t)x_t(1-\phi )$ of polluting firms which meet nonpolluting ones and change type during the time interval because they learn that being nonpolluting is more convenient.

The following formula is then obtained:

(10) \begin{equation} x_{t+1} = x_t\Bigl [ 1 + (1 - x_t)\Bigl ( 2\phi \bigl ( \delta (q^e_{t+1}) \bigr ) - 1 \Bigr ) \Bigr ], \end{equation}

where $\phi \bigl ( \delta (q^e_{t+1}) \bigr )$ represents the probability for a single firm to switch from nonpolluting to polluting. Note that the probability for making the opposite change (switching from polluting to nonpolluting) is then $1 - \phi$ .

To specify function $\phi$ , we introduce the propensity for greenery. More precisely, the probability function $\phi$ , that is, the probability (easiness) with which a nonpolluting firm can become—based on economic convenience—polluting, is assumed to depend on function $\alpha$ measuring the “easiness” with which an honest firm changes its behavior and becomes dishonest. This term allows us to consider different attitudes towards greenery depending on the environmental quality level, that is, when the environmental quality is high and more firms act to fight pollution, then it is less likely for a nonpolluting firm to change type. In addition, as in Bose et al. (Reference Bose, Capasso and Murshid2008) such a function is assumed to be asymmetric, that is, while all polluting firms meeting nonpolluting ones will choose to become honest if the expected profits from honest behavior is not less than the expected profits derived from dishonest behavior, only a fraction of nonpolluting firms meeting polluting ones will choose to change type despite the possibility of higher expected profits being reached. This assumption can be justified by the consideration that a dishonest firm is reluctant to admit its true nature and to spread information about its profit. In this sense, our model allows us to consider a sort of natural propensity for greenery (differently from previous works considering symmetric mechanisms, such as Lamantia and Pezzino (Reference Lamantia and Pezzino2021)).

Hence, $\phi$ is formalized by the following continuous, increasing, and piecewise-smooth function:

(11) \begin{equation} \phi \bigl ( \delta (q^e_t) \bigr ) = \left \{ \begin{array}{l@{\quad}l} 1 - \dfrac {1}{\alpha (E_t) \delta (q^e_t)+1}, & \text{if}\;\; \delta (q^e_t)\geq 0, \\ 0, & \text{if}\;\; \delta (q^e_t)\lt 0, \end{array} \right . \end{equation}

where the function $\alpha (E_t)\geq 0$ is a strictly decreasing continuous and differentiable function such that if $E_t\to 1^-$ then $\alpha (E_t)\to 0^+$ while if $E_t\to 0^+$ then $\alpha (E_t)\to +\infty$ . A possible function presenting such a shape is the following:

(12) \begin{equation} \alpha (E_t)=h\frac {1-E_t}{E_t},\quad h\gt 0, \end{equation}

where parameter $\frac {1}{h}$ represents the strength attached to the propensity for greenery and can be linked to social norms and intrinsic characteristics of countries regarding the sensibility of firms towards environmental issues.

By substituting (12) into (11) and taking into account (6), we obtain the following equation:

(13) \begin{equation} \phi \bigl ( \delta (q^e_t) \bigr ) = \left \{ \begin{array}{l@{\quad}l} \dfrac {h(1-E_t)\bigl [ \Delta _c - (\Delta _c +f) q^e_t \bigr ]}{h(1-E_t)\bigl [ \Delta _c - (\Delta _c +f) q^e_t\bigr ]+E_t}, & \text{if}\;\; q^e_t\leq \bar {q}, \\ 0, & \text{if}\;\; q^e_t\gt \bar {q}, \end{array} \right . \end{equation}

with

(14) \begin{equation} \bar {q} = \dfrac {\Delta _c}{\Delta _c+f} \lt 1. \end{equation}

Using the switching probability (13) in the expression (10) results in the following equation describing the evolution of dishonest firms over time, depending on both the expected monitoring level and the environmental quality in the word-of-mouth mechanism:

(15) \begin{equation} x_{t+1} = S_1(x_t, q^e_t, E_t) \stackrel {df}{=} \left \{ \begin{array}{l@{\quad}l} S_{1,{\mathcal{L}}}(x_t, q^e_t, E_t), & \text{if}\;\; q^e_t\leq \bar {q}, \\ S_{1,{\mathcal{R}}}(x_t, q_t^e, E_t), & \text{if}\;\; q^e_t \gt \bar {q}, \end{array} \right . \end{equation}

where

(16a) \begin{align} S_{1,{\mathcal{L}}}(x_t, q_t^e, E_t) &= x_t + x_t(1 - x_t) \dfrac {h(1 - E_t)\bigl [ \Delta _c-(\Delta _c +f) q^e_t \bigr ] - E_t} {h(1 - E_t) \bigl [\Delta _c - (\Delta _c + f) q^e_t \bigr ] + E_t}, \end{align}

(16b) \begin{align} S_{1,{\mathcal{R}}}(x_t, q_t^e, E_t) &= x_t^2, \end{align}

and $S_{1,{\mathcal{L}}}(x_t, \bar {q}, E_t) = S_{1,{\mathcal{R}}}(x_t, \bar {q}, E_t)$ .

2.5 The piecewise-smooth dynamical system

By taking into account the previous factors, to obtain the final system we need to specify the expectation formation mechanism by the firms on the expected monitoring level. In this work, we consider that a firm knows the rule used by the State to set and update the monitoring level, since it depends on the backward component, that is, the previous monitoring level that is updated taking into account the environmental quality variation, depending on the fraction of polluting firms. Hence, following Coppier et al. (Reference Coppier, Michetti and Scaccia2023), complete information is assumed, that is, the firm knows the monitoring level function used by the State:

\begin{equation*}q^e_t=q_t.\end{equation*}

Different more complex assumptions about the expectation mechanism are left for the future development of the present model.

By combining equations (7), (9), and (15) derived in Sections 2.2–2.4, one obtains a three-dimensional continuous non-smooth map $S\::\: \mathbb{R}^3 \to \mathbb{R}^3$ . In the state space $\mathbb{R}^3$ , the feasible domain for the variables $x_t$ , $q_t$ , and $E_t$ is the unit cube $\mathcal{K} = [0,1]^3$ . In general, for the function $G$ in (9) there is $G \::\: [0, 1] \to [0, 2]$ with the latter limit exceeding the feasible value for $q_t$ . One of the possible workarounds is to introduce the $\max$ function:

(17) \begin{equation} \tilde {G}(x, q, E) \stackrel {df}{=} S_2(x, q, E) = \max \left \{ q \left ( 1 - db(1 - E) + da x \right ), 1 \right \}. \end{equation}

Using the form given in (17) instead of the one defined in (9) influences neither the location of asymptotic states nor their stability properties. It changes only transient parts of some orbits. For particular parameter constellations, the basins of attraction of different attractors may also undergo slight deformations. However, we have not encountered such cases in our numerical simulations.

With the latter update, the evolution of the fraction of polluting firms, monitoring level of the State, and environmental index, is described by the final map $S: \mathcal{K} \ni (x_t, q_t, E_t) \to (x_{t+1}, q_{t+1}, E_{t+1}) \in \mathcal{K}$ such that

(18) \begin{equation} (x_{t+1}, q_{t+1}, E_{t+1}) = S(x_t, q_t, E_t) = \left ( \begin{array}{c} S_1(x_t, q_t, E_t) \\ S_2(x_t, q_t, E_t) \\ S_3(x_t, q_t, E_t) \end{array} \right ) \end{equation}

with $S_1$ given in (15)–(16), $S_2$ in (17), and $S_3$ in (7). The plane $q = \bar {q}$ with the latter defined in (14) divides the cube $\mathcal{K}$ of feasible states into two subdomains, $\mathcal{K}_{{\mathcal{L}}} = [0, 1] \times [0, \bar {q}] \times [0, 1]$ and $\mathcal{K}_{{\mathcal{R}}} = [0, 1] \times (\bar {q}, 1] \times [0, 1]$ .

3.1 Fixed points

As the first step, let us compute the fixed points of the dynamical system (18). Equating

(19) \begin{equation} x = S_1(x, q, E), \quad q = S_2(x, q, E), \quad E = S_3(x, q, E), \end{equation}

and solving for $x$ , $q$ , and $E$ , we derive the following proposition that trivially holds.

Proposition 3.1 implies that system $S$ has infinitely many fixed points. The points $P_0^q$ and $P_1^q$ are always feasible for $q \in [0, 1]$ . The former are referred to as good equilibria, as they are associated with the situation with no polluting firms and maximum environmental quality level, while the latter are called bad equilibria, since all firms pollute and the environmental quality is at the minimum level. The presence of these two kinds of stationary points is clearly due to the word-of-mouth mechanism: if all firms are of the same type, they cannot meet firms of different types, and hence they have no opportunities to obtain new information and change their behavior.

In Figure 1 we plot the feasible cube $\mathcal{K}$ together with the fixed points of different types. The good and the bad equilibria are shown with dark blue and dark red, respectively.

Figure 1. The cube $\mathcal{K}$ of feasible values together with the fixed points $P_0^q$ (dark blue), $P_1^q$ (dark red), and $P_{\ast }^E$ (dark green). The parameters are $\Delta _c = 1, f = 1, d = 0.9, b = 0.9$ and (a) $h = 0.35, a = 0.6$ ; (b) $h = 1, a = 0.85$ .

While our model always admits the good and bad equilibria, a question to be addressed refers to the existence of the interior fixed points depending on the parameters of the system. More precisely, the curve $\mathcal{L}_{\ast }$ defined by (21), consisting of the nontrivial fixed points $P_{\ast }^E$ , $E \in [0, 1]$ , can have the empty intersection with the cube $\mathcal{K}$ . Indeed,

(22) \begin{equation} P_{\ast }^0 = \left ( \dfrac {b}{a}, \bar {q}, 0 \right ) \quad \text{and} \quad P_{\ast }^1 = (0, -\infty , 1). \end{equation}

The point $P_{\ast }^0$ is outside $\mathcal{K}$ , since $\frac {b}{a} \gt 1$ , as well as the limit $P_{\ast }^1$ does not belong to $\mathcal{K}$ . Moreover, both functions $x^\ast (E)$ and $q^\ast (E)$ are strictly decreasing, since

\begin{equation*} \dfrac {\mathrm{d} x^\ast (E)}{\mathrm{d} E} = -\dfrac {b}{a} \lt 0 \quad \text{and} \quad \dfrac {\mathrm{d} q^\ast (E)}{\mathrm{d} E} = -\dfrac {1}{h(1 - E)^2(\Delta _c + f)} \lt 0. \end{equation*}

It means that $x^\ast (E)$ and $q^\ast (E)$ decrease along $\mathcal{L}_{\ast }$ with increasing $E$ . Then the condition for having a non-empty intersection $\mathcal{L}_{\ast } \cap \mathcal{K} \neq \varnothing$ is that there exist $E_{\text{A}} \lt E_{\text{B}}$ defined by $x^\ast (E_{\text{A}}) = 1$ and $q^\ast (E_{\text{B}}) = 0$ . The fragment of $\mathcal{L}_{\ast }$ between the points

\begin{equation*} P_{\ast }^{E_{\text{A}}}\Bigl (x^\ast (E_{\text{A}}), q^\ast (E_{\text{A}}), E_{\text{A}}\Bigr ) \quad \text{and} \quad P_{\ast }^{E_{\text{B}}}\Bigl (x^\ast (E_{\text{B}}), q^\ast (E_{\text{B}}), E_{\text{B}} \Bigr ) \end{equation*}

is located inside $\mathcal{K}$ . Solving (21) for $E_{\text{A}}$ and $E_{\text{B}}$ and using the aforementioned conditions, one obtains

(23) \begin{align} x^\ast (E_{\text{A}}) &= 1, & q^\ast (E_{\text{A}}) &= \bar {q} - \dfrac {b - a}{ah(\Delta _c + f)} \stackrel {df}{=} q_{\text{A}}, & E_{\text{A}} &= E_{\mathrm{bad}}, \end{align}

(24) \begin{align} x^\ast (E_{\text{B}}) &= \dfrac {b}{a(\Delta _c h + 1)} \stackrel {df}{=} x_{\text{B}}, & q^\ast (E_{\text{A}}) &= 0, & E_{\text{B}} &= \dfrac {\Delta _c h}{\Delta _c h + 1}, \end{align}

and consequently

(25) \begin{equation} P_{\ast }^{E_{\text{A}}} = \left ( 1, q_{\text{A}}, E_{\mathrm{bad}} \right ) \quad \text{and} \quad P_{\ast }^{E_{\text{B}}} = \left ( x_{\text{B}}, 0, E_{\text{B}} \right ). \end{equation}

Note that $q_{\text{A}} \lt \bar {q}$ and the feasible $P_{\ast }^E$ , if it exists, always belongs to $\mathcal{K}_{{\mathcal{L}}}$ . In terms of the parameters, the condition for $E_{\text{A}} \lt E_{\text{B}}$ is

(26) \begin{equation} b \lt a \left ( \Delta _c h + 1 \right ), \end{equation}

which also implies $x_{\text{B}} \lt 1$ and $q_{\text{A}} \gt 0$ . Hence, provided that (26) holds, the points $P_{\ast }^E \in \mathcal{K}_{{\mathcal{L}}}$ with $E_{\text{A}} \leq E \leq E_{\text{B}}$ represent feasible fixed points and are referred to as internal equilibria. The other points $P_{\ast }^E \not \in \mathcal{K}_{{\mathcal{L}}}$ with $E \in [0, 1] \setminus [E_{\text{A}}, E_{\text{B}}]$ are non-feasible. The abovementioned arguments prove the following proposition.

Proposition 3.2. Consider the map $S$ defined in (7) and (15)–(18). If

\begin{equation*}b \lt a \left ( \Delta _c h + 1 \right ),\end{equation*}

then the interior equilibria are feasible.

In Figure 1 the curve $\mathcal{L}_{\ast }$ is shown in dark green. The projections of $\mathcal{L}_{\ast }$ (more precisely, of its part visible in the figure) onto the planes $x = 0$ , $q = 1$ , and $E = 0$ are colored light green. In the panel (a), there is $E_{\text{B}} \lt E_{\text{A}}$ (i.e., (26) does not hold). It means that $\mathcal{L}_{\ast } \cap \mathcal{K} = \varnothing$ and there are no feasible nontrivial equilibria. On the contrary, in the panel (b), one observes part of the curve $\mathcal{L}_{\ast }$ located between the points $P_{\ast }^{E_{\text{A}}}$ and $P_{\ast }^{E_{\text{B}}}$ , which represents the internal equilibria of the map $S$ .

Note that the fixed points characterized by the coexistence of polluting and nonpolluting firms are owned as long as condition (26) holds. Regarding the interior equilibrium $P^E_*$ , we can observe that for all environmental quality levels, the equilibrium fraction of polluting firms $x^*$ depends negatively on the environmental quality $E$ , as can be seen from the partial derivative. This negative relationship between the fraction of non-compliant firms and environmental quality passes through the propensity for greenery: in fact, ceteris paribus, the evolution of dishonest firms over time depends on environmental quality through the mechanism of word-of-mouth, that is, as the environmental quality increases, the probability that a green firm changes into polluting one decreases. On the other hand, regarding the equilibrium monitoring level $q^*$ , if the environmental quality improves, all things being equal, the level of monitoring by the State is reduced. Furthermore, it can be shown that the level of internal equilibrium monitoring has an inverse relationship with the propensity for greenery $\frac {1}{h}$ : if the State were able to increase the attention of firms to the environment, it could set a lower level of control on the dishonesty of firms.

As described, the system can admit up to three different types of coexisting fixed points, so the investigation of their stability becomes crucial in order to establish the system’s attractor from a qualitative-quantitative point of view for different parameter values and initial conditions and to give economic insight.

3.2 Stability

In this section, we consider the question concerning the stability of fixed points. Clearly, for every fixed point mentioned in Proposition 3.1, inside its arbitrarily small neighborhood there is an uncountable number of other fixed points. Hence, every such fixed point is always locally (neutrally) stable at least in one direction. In the other directions, these points can be stable or unstable, depending on the parameters and the location of the point itself.

Let us consider first the good equilibria $P_0^q$ , representing desirable asymptotic states, where there are no dishonest firms and the environment level is the highest possible. Not depending on whether $q \leq \bar {q}$ or $q \gt \bar {q}$ , the respective Jacobian is

(27) \begin{equation} J_{{\mathcal{L}}}\left ( P_0^q \right ) \equiv J_{{\mathcal{R}}}\left ( P_0^q \right ) = \left ( \begin{array}{c@{\quad}c@{\quad}c} 0 & 0 & 0 \\ qda & 1 & qdb \\ -a & 0 & 1-b \end{array} \right ), \end{equation}

the eigenvalues of which are $\lambda _1^{\text{good}} = 0$ , $\lambda _2^{\text{good}} = 1$ , $\lambda _3^{\text{good}} = 1 - b$ , $0 \lt \lambda _3^{\text{good}} \lt 1$ . The following proposition trivially holds.

According to the previous result, there exists an environmental level $\bar {E}$ and a fraction of dishonest firms $\bar {x}$ such that if the economy starts from a situation in which the environmental quality is high enough (i.e., $E_0\gt \bar {E}$ ) and the initial fraction of dishonest firms is low enough (i.e., $x_0\lt \bar {x}$ ), then the good equilibrium will be reached in the long term for all given parameter constellations.

Furthermore, we consider the bad equilibria $P_1^q$ , which are not desirable to be reached, since all firms switch to dishonest behavior. However, the environmental level can be rather high in the case when $a \ll b$ . If $q \leq \bar {q}$ , the respective Jacobian is

(28) \begin{equation} J_{{\mathcal{L}}}\left ( P_1^q \right ) = \left ( \begin{array}{c@{\quad}c@{\quad}c} \dfrac {2(b - a)}{b - a\left ( 1 - h \left ( \Delta _c + f \right )(\bar {q} - q) \right )} & 0 & 0 \\ qda & 1 & qdb \\ -a & 0 & 1-b \end{array} \right ) \end{equation}

and its eigenvalues are $\lambda _1^{\text{bad}} = 1$ , $\lambda _2^{\text{bad}} = 1 - b$ , and

(29) \begin{equation} \lambda _3^{\text{bad}} = \lambda _3^{\text{bad}}(q) = \dfrac {2(b - a)} {b - a + ah\left ( \Delta _c + f \right )(\bar {q} - q)}. \end{equation}

If $q \gt \bar {q}$ , the respective Jacobian is

(30) \begin{equation} J_{{\mathcal{R}}}\left ( P_1^q \right ) = \left ( \begin{array}{c@{\quad}c@{\quad}c} 2 & 0 & 0 \\ qda & 1 & qdb \\ -a & 0 & 1-b \end{array} \right ) \end{equation}

the eigenvalues of which are $\lambda _1^{\text{bad}} = 1$ , $\lambda _2^{\text{bad}} = 1 - b$ , $\lambda _3^{\text{bad}} = 2$ . The following proposition holds.

Proof. For $q \gt \bar {q}$ a fixed point $P_1^q$ is clearly unstable. Consider $q \leq \bar {q}$ . A point $P_1^q$ is locally stable if $\lambda _3^{\text{bad}}$ given by (29) is between $-1$ and $+1$ . Taking the derivative

(31) \begin{equation} \dfrac {\mathrm{d} \lambda _3^{\text{bad}}}{\mathrm{d} q} = \dfrac {2ah(b - a)(\Delta _c + f)}{\Bigl ( b - a + ah\left ( \Delta _c + f \right )(\bar {q} - q) \Bigr )^2} \gt 0, \quad q \neq \bar {q} + \dfrac {b - a}{ah(\Delta _c + f)}, \end{equation}

one deduces that $\lambda _3^{\text{bad}}(q)$ is an increasing function almost everywhere. From (29) it follows that this function is positive-valued for

(32) \begin{equation} q \lt q_{\mathrm{as}} = \bar {q} + \dfrac {b - a}{ah(\Delta _c + f)}. \end{equation}

Since $q_{\mathrm{as}} \gt \bar {q}$ , the fixed point $P_1^q$ for $q \leq \bar {q}$ can become unstable only due to $\lambda _3^{\text{bad}}(q)$ passing through $+1$ . The latter occurs when $q = q_{\text{A}}$ .

The bad equilibrium can be stable or unstable, and, according to the previous proposition, sufficient conditions for $P^q_1$ being locally unstable is given by $q\geq q_A$ . Therefore, for the bad equilibrium to be stable, the level of monitoring of firms must be low; conversely, if the State were to set a higher monitoring level, it would make the bad equilibrium unstable and thus move the economy away from an undesirable equilibrium.

Recall also that if $b\gt a(\Delta _ch+1)$ (i.e., (26) does not hold), then no nontrivial (internal) fixed point $P_{\ast }^E$ exists. The latter also means $q_{\text{A}} \lt 0$ , so that all bad equilibria $P_1^q$ , $q \in [0, 1]$ , are unstable. According to the previous considerations, we can obtain conditions on parameters guaranteeing that almost all initial conditions converge towards the good equilibrium. The following remark then holds.

Note that the condition of the Remark 3.5 can also be written in terms of the propensity for greenery in order to give a better economic insight into the meaning of this condition:

(33) \begin{equation} \frac {1}{h} \gt \frac {a \Delta _c}{b-a} \stackrel {df}{=} \frac {1}{\bar {h}}. \end{equation}

Thus, the set of good equilibria is globally asymptotically stable if the propensity for greenery $\frac {1}{h}$ is high enough, or the difference in the costs of green and nongreen technology is small. Clearly, both represent strategies that are expensive to implement and also take a long time. A State could, therefore, try to put the economy on a virtuous path by, for example, “educating” the population towards greening and/or at the same time investing or encouraging investments that make green technology more economically viable.

Finally, we consider the feasible internal equilibria $P_{\ast }^E$ , $E \in [E_{\text{A}}, E_{\text{B}}]$ , when they exist, that is $E_{\text{A}} \lt E_{\text{B}}$ . The respective Jacobian is

(34) \begin{equation} J_{{\mathcal{L}}}\left ( P_{\ast }^E \right ) = \left ( \begin{array}{c@{\quad}c@{\quad}c} 1 & h(\Delta _c + f) G (1 - E)^2 & G \\ Ha & 1 & Hb \\ -a & 0 & 1-b \end{array} \right ) \end{equation}

with

(35) \begin{equation} G = \dfrac {b \Bigl ( b(1 - E) - a \Bigr )}{2Ea^2}, \quad H = \dfrac {d \Bigl ( \Delta _c h (1 - E) - E \Bigr )}{h(\Delta _c + f)(1 - E)}. \end{equation}

Its eigenvalues are $\lambda _1^{\mathrm{int}}(E) = 1$ and

(36) \begin{equation} \lambda _{2,3}^{\mathrm{int}}(E) = 1 - \dfrac {b}{2} \pm \dfrac {1}{2} \sqrt {\frac {2b\Bigl ( b(1 - E) - a \Bigr )}{Ea} \biggl ( d\Bigl ( \Delta _c h(1 - E) - E \Bigr )(1 - E) - 1 \biggr ) + b^2}. \end{equation}

The condition in Proposition 3.6 does not allow for the explicit solution in the general case. However, certain comments can be made concerning whether $\lambda _{2,3}^{\mathrm{int}}(E)$ are real or complex. Indeed, the discriminant (the expression under the square root) of (36) can be rewritten as

(37) \begin{equation} D = \dfrac {bQ(E)}{Ea}, \quad Q(E) = 2 ( b(1 - E) - a ) ( d ( \Delta _c h(1 - E) - E )(1 - E) - 1 ) + baE. \end{equation}

In case $Q(E) \lt 0$ , the eigenvalues $\lambda _{2,3}^{\mathrm{int}}(E)$ are complex and the corresponding fixed point is a focus. The function $Q(E)$ is a cubic polynomial with respect to $E$ , which can have from one to three real roots. Whether the (feasible) internal equilibria $P_{\ast }^E$ are nodes/saddles or foci depends on the location of these roots with respect to $[E_{\text{A}}, E_{\text{B}}]$ . Let us introduce the point of minimum

(38) \begin{equation} E_{+} \in (0, 1) \::\: Q(E_{+}) = \min _{E \in (0, 1)} Q(E). \end{equation}

The following statement holds.

Proof. We notice that

\begin{equation*} Q(E_{\text{A}}) = a(b - a) \gt 0 \quad \Rightarrow \quad \lambda _{2}^{\mathrm{int}}(E_{\text{A}}) = 1,\;\; \lambda _{3}^{\mathrm{int}}(E_{\text{A}}) = 1 - b. \end{equation*}

Let us check the sign of $Q(E_{\text{B}})$ . We have

\begin{equation*} Q(E_{\text{B}}) \lt 0 \quad \Leftrightarrow \quad \Delta _c h a (b + 2) \lt 2(b - a) \quad \Leftrightarrow \quad h \lt \dfrac {2(b - a)}{\Delta _c a (b + 2)}. \end{equation*}

However,

\begin{equation*} \dfrac {2(b - a)}{\Delta _c a (b + 2)} \lt \dfrac {b - a}{a\Delta _c} \quad \Leftrightarrow \quad 2 \lt b + 2, \end{equation*}

which is always true. Then in the case when feasible $P_{\ast }^E$ exist, there is always $Q(E_{\text{B}}) \gt 0$ . Let us check the possibility of $Q(E) \lt 0$ for $E \in (E_{\text{A}}, E_{\text{B}})$ .

Collected with respect to $E$ , the polynomial $Q(E)$ becomes

(39) \begin{equation} Q(E) = c_3E^3 + c_2 E^2 + c_1 E + c_0 \end{equation}

with

(40) \begin{align} c_0 &= 2(d\Delta _c h - 1)(b - a), & c_1 &= -2\Delta _c h d(3b - 2a) - 2d(b-a) + b(a + 2), \nonumber \\ c_2 &= 2d\Bigl ( \Delta _c h(3b - a) + 2b - a\Bigr ), & c_3 &= -2bd(\Delta _c h + 1), \end{align}

and moreover, $c_2 \gt 0$ , $c_3 \lt 0$ . The derivative

\begin{equation*} Q^{\prime }(E) = 3c_3 E^2 + 2 c_2 E + c_1 = 0 \end{equation*}

results in two extrema points

(41) \begin{equation} E_{\pm } = \dfrac {-c_2 \pm \sqrt {c_2^2 - 3c_1c_3}}{3c_3}, \end{equation}

which always exist. Moreover, there is $E_{-} \gt 1$ , $E_{+} \lt E_{\text{B}}$ , and $Q^{\prime \prime }(E_{+}) \gt 0$ . Thus, $E_{+}$ is the minimum point, and hence $E_{-}$ is the maximum point (for the detailed proof concerning the properties of $E_{\pm }$ see Appendix A). From this, the statement of the proposition follows directly. Indeed, if $E_{+} \leq E_{\text{A}}$ , the function $Q(E)$ is increasing in $[E_{\text{A}}, E_{\text{B}}]$ , and cannot become negative. Neither can $Q(E)$ become negative if $E_{+} \in (E_{\text{A}}, E_{\text{B}})$ and $Q(E_{+}) \gt 0$ . This implies the case 1. Clearly, if $E_{+} \in (E_{\text{A}}, E_{\text{B}})$ with $Q(E_{+}) \lt 0$ , the case 2 follows.

As already underlined, the interior equilibrium can be locally stable or it can be a saddle point; when stable the convergence may be monotonic or oscillating. In addition, as more than one kind of coexisting attractors can be owned, the study of their basins becomes crucial. To attach such questions and to give economic insight, we will use numerical simulations illustrating different dynamic situations.

3.3 Numerical simulations

To better understand the possible features of the system and to give economic insight, in what follows we distinguish between the following cases:

1. (1) the map $S$ has only stable good equilibria $P_0^q$ ;

2. (2) the map $S$ has both stable good and stable bad equilibria $P_0^q$ and $P_1^q$ ;

3. (3) besides stable $P_0^q$ and $P_1^q$ the map $S$ also admits stable nodes or stable foci $P_{\ast }^E$ .

3.3.1 All stable fixed points are good equilibria

If $b \gt a(\Delta _c h + 1)$ , which is equivalent to

(42) \begin{equation} \frac {1}{h} \gt \frac {1}{\bar {h}}, \end{equation}

the internal equilibria $P_{\ast }^E$ are non-feasible. As stated in Remark 3.5, almost all initial conditions produce orbits converging to the good equilibria $P_0^q$ . This case is related to high propensity for greenery.

This situation is presented in Figure 2 where we plot the 3D phase space fixing $\Delta _c = 1, a = 0.85, b = 0.9$ , $f = 1$ , $d = 0.9$ , then, since $\frac {b - a}{\Delta _c a} \approx 0.0588$ , we put $h = 0.05$ so that condition (42) is satisfied. We fix the initial $E_0^1 = 0.1$ and $E_0^2 = 0.9$ , and check several combinations of $x_0$ and $q_0$ : $(0.2, 0.1)$ , $(0.7, 0.4)$ , $(0.9, 0.9)$ . The two orbits are shown with dark red ( $E_0 = E_0^1$ ) and dark blue ( $E_0 = E_0^2$ ), while by pink and cyan, respectively, the projections of these orbits onto the planes $x = 0$ , $q = 1$ , and $E = 0$ are plotted.

Figure 2. The orbits for (a) $ x_0 = 0.2, q_0 = 0.1$ , (b) $x_0 = 0.7, q_0 = 0.4$ , (c) $x_0 = 0.9, q_0 = 0.9$ . Dark red color corresponds to $E_0 = E_0^1 = 0.1$ and dark blue to $E_0 = E_0^2 = 0.9$ . With pink and cyan colors the respective projections onto $x = 0$ , $q = 1$ , and $E = 0$ are shown. The parameters are $\Delta _c = 1, f = 1, d = 0.9, h = 0.05, a = 0.85, b = 0.9$ .

As already highlighted, if the propensity for greenery is high enough or $\Delta _c$ is low enough, then almost all initial conditions produce orbits converging to a good fixed point, and these represent sufficient conditions. However, to achieve a large propensity for greenery or making the cost difference between green and nongreen technologies being minimal, both scenarios indicate strategies that are costly and time-consuming to implement. Consequently, the State might aim to set the economy on a sustainable trajectory by, for instance, raising public awareness on the benefits of green technology and/or actively investing in or encouraging investments that make green technology more economically competitive. However, this requires high economic resources and a medium/long term time horizon, which is often not compatible with political horizons that are short term.

3.3.2 Stable fixed points can be both good or bad equilibria

If $b \lt a(\Delta _c h + 1)$ , which is equivalent to

(43) \begin{equation} \frac {1}{h} \lt \frac {1}{\bar {h}}, \end{equation}

there are feasible $P_{\ast }^E$ and stable $P_1^q$ . In this case, if the propensity for greenery is not too small, then the interior equilibria are saddle points, so we observe the coexistence of two kinds of equilibria, the good and the bad ones, and the study of their basins becomes interesting. Such a situation emerges if the propensity for greenery is at an intermediate level.

Let us first consider the case, when all $P_{\ast }^E$ have only real eigenvalues. Keeping $\Delta _c = 1, a = 0.85, b = 0.9, f = 1, d = 0.9$ , we derive that there are no foci for $0.0588 \lessapprox h \lessapprox 2.7$ . So, we fix first $h = 1$ . In this case, all feasible internal equilibria are saddles; hence, almost all orbits are attracted to either $P_0^q$ or $P_1^q$ . In Figure 3 we plot two sections $E_0 = 0.1$ (a) and $E_0 = 0.9$ (b) of the 3D phase space. Here the light blue color denotes the basin of $P_0^q$ , while the gray color corresponds to the basin of $P_1^q$ . In the panel (c) two orbits are plotted, the one ending up at a good equilibrium (dark blue) and the other at a bad equilibrium (dark red). As before, the pink and cyan colors show the projections of these orbits onto the planes $x = 0$ , $q = 1$ , and $E = 0$ .

Figure 3. The sections (a) $E_0 = 0.1$ and (b) $E_0 = 0.9$ of the 3D phase space $\mathcal{K}$ . The basins of $P_0^q$ and $P_1^q$ are filled with light blue and gray, respectively. In (c) two orbits are plotted, for $x_0 = 0.8, q_0 = 0.3$ and $E_0 = 0.1$ (dark red) and $E_0 = 0.9$ (dark blue). With the pink and cyan colors the respective projections onto $x = 0$ , $q = 1$ , and $E = 0$ are shown. The parameters are $\Delta _c = 1, f = 1, d = 0.9, h = 1, a = 0.85, b = 0.9$ .

As it can be observed, $\exists x_m$ such that if $x_0\lt x_m$ then the system will converge to a good fixed point. Such evidence suggests that, when the propensity for greenery is not high enough, an intervention to fight dishonest behavior aiming at reducing the initial level of the fraction of dishonest firms can move the system toward a condition converging to a desirable situation, that is, with environmental quality at the maximum level associated to all firms being compliant with the rules.

To better understand such evidence, we consider the influence of changing the fine level $f$ and the responsiveness of monitoring changes in environmental quality $d$ . We computed the relative parts of basins of $P_0^q$ and $P_1^q$ fixing three pairs $(f, d)$ : (a) $f = 1, d = 0.9$ ; (b) $f = 3, d = 0.9$ ; (c) $f = 1, d = 0.4$ . In (a) the basins of $P_0^q$ and $P_1^q$ constitute 87.19 and 12.81% of $\mathcal{K}$ , respectively; in (b) 92.99 and 7.01%; and in (c) 86.93 and 13.07%. Consequently, by increasing $f$ the basin of good equilibria becomes bigger and the one of bad equilibria shrinks, while the effect of decreasing $d$ seems to be ambiguous.

In Figure 4 for the three mentioned parameter pairs, we plot the section $E_0 = 0.7$ of the feasible cube $\mathcal{K}$ , where the basins of the good and the bad equilibria are filled with light blue and gray colors, respectively. In panel (b), the basin of $P_1^q$ is significantly smaller than in panels (a) and (c), although the difference between panels (a) and (c) is not so evident. Thus, in the case in which the propensity for greenery is at an intermediate level, the fine envisaged for non-compliant firms which are discovered, can be an effective tool to reduce the likelihood of converging to a bad state.

Figure 4. The section $E_0 = 0.7$ of the 3D phase space $\mathcal{K}$ for (a) $f = 1, d = 0.9$ , (b) $f = 3, d = 0.9$ , (c) $f = 1, d = 0.4$ . The basins of $P_0^q$ and $P_1^q$ are filled with light blue and gray, respectively. The other parameters are $\Delta _c = 1, h = 1, a = 0.85, b = 0.9$ .

Figure 5. The time series of $x$ , $q$ , and $E$ for the two orbits with $d = 0.1$ (dark red) and $d = 0.9$ (dark blue). The initial conditions are (a)–(c) $x_0 = 0.9$ , $q_0 = 0.4$ , $E_0 = 0.3$ , and (d)–(f) $x_0 = 0.9$ , $q_0 = 0.2$ , $E_0 = 0.9$ . The other parameters are $\Delta _c = 1, f = 1, h = 1, a = 0.85, b = 0.9$ .

While the influence of the parameter $f$ is evident, the effect of $d$ does not seem to be very clear. To better understand the role of the parameter $d$ , we plot in Figure 5 the time series of two orbits with the same initial conditions but obtained for different values of $d$ , namely, the low $d = 0.1$ and the high $d = 0.9$ . In the panels (a), (c) we show the evolution of $x$ and $E$ of two orbits converging to a good equilibrium. As one can see, decreasing $d$ decreases the speed of convergence to the asymptotic state. The panels (d), (f) present the same time series for two orbits converging to a bad equilibrium. Now, with a smaller $d$ the orbit converges to the fixed point quicker. Thus, the reactivity of monitoring changes in the environment plays a positive role: in fact, a high value of $d$ increases the speed at which the system converges to a good state, while it reduces the speed at which it converges to a bad state. Note that the two orbits shown in the same panel of Figure 5, converge, strictly saying, to different fixed points, with different final levels of $q$ , as shown in panels (b) and (e), where we plot the variable $q_t$ vs. $t$ with the same low and high $d$ . In panel (b), related to good equilibria, with smaller $d$ the State must put more effort into discovering dishonesty (the final $q$ is larger). In panel (e), for bad equilibria the influence of $d$ is the opposite.

3.3.3 Coexistence of three kinds of fixed points, good, bad, and interior equilibria

If the propensity for greenery is low, then the interior equilibria are locally stable and they can be both nodes and foci.

We start with considering the case in which the interior equilibria are stable nodes. With respect to the previous example, we only increase $h = 2.5$ . As before all internal equilibria have only real eigenvalues, but now $P_{\ast }^E$ are stable nodes for $E_{\text{A}} \lt E \lessapprox 0.276$ . In Figure 6 we plot the section $E_0 = 0.4$ of the 3D phase space for different pairs of $f$ and $d$ , where the light blue color denotes the basin of $P_0^q$ , the gray color corresponds to the basin of $P_1^q$ , and violet is associated with $P_{\ast }^E$ .

Figure 6. The section $E_0 = 0.4$ of the 3D phase space $\mathcal{K}$ for (a) $f = 1, d = 0.9$ , (b) $f = 3, d = 0.9$ , (c) $f = 1, d = 0.7$ . The basins of $P_0^q$ , $P_1^q$ , and $P_{\ast }^E$ are filled with light blue, gray, and violet, respectively. The other parameters are $\Delta _c = 1, h = 2.5, a = 0.85, b = 0.9$ .

We again computed the relative parts of basins of $P_0^q$ , $P_1^q$ , and $P_{\ast }^E$ fixing (a) $f = 1, d = 0.9$ ; (b) $f = 3, d = 0.9$ ; (c) $f = 1, d = 0.7$ . In (a) the basins of $P_0^q$ , $P_1^q$ , and $P_{\ast }^E$ constitute 76, 23.43, and 0.57% of $\mathcal{K}$ , respectively; in (b) 87.36, 12.36 and 0.28%; in (c) 76.15, 23.71, and 0.13%. Hence, in this case with the increase of $f$ and the decrease of $d$ , the basins of bad and internal equilibria shrink, while the basin of good equilibria becomes bigger. Note that in the $(d, h)$ parameter plane, the curve $h = \xi (d)$ corresponding to the appearance of stable internal equilibria has the shape of a hyperbola. So, with a fixed $h$ the decrease of $d$ leads at some point to the complete elimination of the basin of internal equilibria.

We then move to the case in which the interior equilibria can be both stable nodes and stable foci. With this aim in mind, we fix $\Delta _c = 1, d = 0.9, f = 1, a = 0.85, b = 0.9$ , and $h = 7$ . For such a parameter constellation, the points $P_{\ast }^E$ with $E_{\text{A}} \lt E \lessapprox 0.061$ or $0.48 \lessapprox E \lessapprox 0.56$ are stable nodes, with $0.061 \lessapprox E \lessapprox 0.107$ or $0.242 \lessapprox E \lessapprox 0.48$ are stable foci, with $0.107 \lessapprox E \lessapprox 0.242$ are unstable foci, while with $0.56 \lessapprox E \lt E_{\text{B}}$ are saddles. In Figure 7(a) we plot the section $E_0 = 0.4$ of the 3D phase space, where light blue denotes the basin of $P_0^q$ , gray corresponds to the basin of $P_1^q$ , while violet and pink are associated with $P_{\ast }^E$ being nodes and foci, respectively. Panel (b) is the enlargement of the boxed area in panel (a). In Panel (c) we show four different orbits ending at a good equilibrium (dark blue), a bad equilibrium (dark red), an internal stable node (violet), and an internal stable focus (dark green). The projections of these orbits to the planes $x = 0$ , $q = 1$ , and $E = 0$ are shown in cyan, pink, lilac, and light green, respectively.

Figure 7. (a) The section $E_0 = 0.4$ of the 3D phase space $\mathcal{K}$ ; (b) is the magnification of the boxed area in (a). The basins of $P_0^q$ , $P_1^q$ , $P_{\ast }^E$ -nodes and $P_{\ast }^E$ -foci are filled with light blue, gray, violet, and pink, respectively. In (c) four orbits are plotted, for $x_0 = 0.6, E_0 = 0.4$ and $q_0 = 0.3$ (dark red), $q_0 = 0.42$ (dark green), $q_0 = 0.47$ (violet), and $q_0 = 0.8$ (dark blue). With pink, light green, lilac, and cyan colors the respective projections onto $x = 0$ , $q = 1$ , and $E = 0$ are shown. The red and the yellow points mark the final $P_{\ast }^E$ -focus and $P_{\ast }^E$ -node, respectively. The parameters are $\Delta _c = 1, f = 1, d = 0.9, h = 7, a = 0.85, b = 0.9$ .

Comparing Figure 6 with Figure 7, one can see that when sensitivity to the environment decreases further, the pool of internal equilibria increases. Furthermore, similarly to the case in which interior fixed points are nodes, increasing $f$ and decreasing $d$ implies the enlargement of the basin of good equilibria. Also with the increase of $f$ , both the basins of bad and internal equilibria shrink. Then with the decrease of $d$ the basin of bad equilibria may become bigger, but the basin of internal equilibria may disappear.