Stability and Bifurcations in a Discrete-Time Eco-Evolutionary Logistic Model

Abstract

In this paper I study a two-dimensional discrete-time evolutionary logistic-type model describing the coupled dynamics of population density and a continuously evolving trait. I provide a local bifurcation analysis of the equilibria, deriving explicit conditions for their existence and local stability. In particular, I show that the boundary and interior equilibria exchange stability through a transcritical bifurcation, and I characterize analytically the subsequent loss of stability of the interior equilibrium via period-doubling and Neimark–Sacker bifurcations. Transversality is established in all cases, and the criticality of the bifurcations is determined through normal form and Lyapunov coefficient computations. I show that the period-doubling bifurcation can be supercritical or subcritical, while the Neimark–Sacker bifurcation is generically nondegenerate and may be either supercritical or subcritical, depending on parameter values.

1. Introduction

Discrete-time eco-evolutionary models have been widely used to investigate the interplay between population dynamics and trait evolution, particularly in settings with non-overlapping generations and density-dependent regulation [1]. A growing body of work has shown that incorporating evolutionary feedback into classical population models can fundamentally alter their dynamical behavior, leading to rich phenomena such as multistability, oscillations, and bifurcations that are absent in purely ecological formulations [2,3,4,5,6,7]. In this context, several recent studies have explored evolutionary extensions of discrete-time growth models, focusing on stability, bifurcation structure, and the emergence of complex dynamics in canonical population models of Ricker type [8,9,10,11,12] and in the Beverton–Holt model [13,14,15,16,17], highlighting the generality and robustness of eco-evolutionary feedback mechanisms across different density-dependent growth formulations.

Among these models, the one-dimensional logistic map occupies a paradigmatic position [18,19,20,21]. Despite its apparent simplicity, it captures a rich variety of dynamical regimes and has become a canonical example in the study of discrete dynamical systems, both as a research tool and as a pedagogical model [20]. Its systematic analysis has shaped much of the modern understanding of bifurcation theory, chaos, and nonlinear dynamics [18].

Motivated by recent developments in eco-evolutionary theory, there has been growing interest in extending such classical population models to incorporate evolutionary processes and feedback mechanisms [2,4,5,6]. In particular, discrete-time eco-evolutionary models allow one to study how trait evolution interacts with population regulation, potentially giving rise to novel dynamical behaviors that are absent in purely ecological systems. However, the inclusion of evolutionary variables typically increases both the dimensionality and the analytical complexity of the model, making rigorous analysis substantially more challenging [4].

Despite the extensive literature on eco-evolutionary extensions of classical discrete-time growth models, an evolutionary counterpart of the logistic map has not been developed in a unified and systematic way. While evolutionary versions of Ricker-type and Beverton–Holt models have been introduced and rigorously analyzed in recent years, an analogous extension for the logistic population model remains comparatively unexplored. The main motivation of this paper is to fill this gap by proposing and studying a discrete-time eco-evolutionary system whose ecological component preserves logistic-type density regulation, thereby providing a natural evolutionary extension of the classical logistic map.

Thus, in this work, I study a two-dimensional discrete-time eco-evolutionary system in which population dynamics are governed by a logistic-type mechanism, while trait evolution is driven by stabilizing selection and nonlinear eco-evolutionary feedback. The model can be viewed as a natural extension of the classical logistic map, retaining its bounded ecological phase space while enriching the dynamics through evolutionary coupling. This structure makes the system particularly well suited for a systematic and rigorous dynamical analysis.

Throughout this paper, I use standard tools from discrete dynamical systems, including invariant set analysis, linearization, Jury stability criteria, center manifold reduction, and normal form theory [22,23]. Using these methods, I provide a complete description of the existence and stability of equilibria, establish the presence of a compact global attractor, and rigorously classify the bifurcations of the interior equilibrium, including transcritical, period-doubling, and Neimark–Sacker bifurcations.

From a biological viewpoint, $x\left(t\right)$ represents the (scaled) population density of a species with non-overlapping generations, while $u\left(t\right)$ represents a continuous quantitative trait (e.g., the trait mean) affecting fitness. The factor $exp(-u{\left(t\right)}^2/2)$ models stabilizing selection around an optimal trait value (set to 0 without loss of generality), so that deviations from the optimum reduce per capita reproduction. The parameter ${\sigma }^2$ can be interpreted as an effective genetic variance (or, more broadly, evolutionary responsiveness), controlling the rate at which the trait responds to selection. The feedback term $x/(1-c_{0}x)$ captures density-dependent selection pressures mediated by crowding or resource limitation; in particular, it increases sharply as x approaches $1/c_0$, thereby strengthening the eco-evolutionary coupling at high densities.

Overall, the model provides a minimal benchmark for eco-evolutionary dynamics in seasonal or discrete-generation populations (e.g., annual plants, insects, or laboratory populations), where demography and rapid trait change may interact on comparable timescales.

Beyond its specific ecological interpretation, this work is also motivated by the need for analytically tractable benchmark models in discrete-time eco-evolutionary theory. In the same spirit in which the one-dimensional logistic map is often used as a first introduction to nonlinear dynamics, the present model provides a natural next step: it retains a logistic-type ecological backbone while incorporating biologically meaningful evolutionary feedback, and it remains amenable to a systematic, rigorous analysis. I therefore expect it to serve as a pedagogical case study illustrating how classical techniques extend to low-dimensional eco-evolutionary systems and to be useful both for researchers entering the field and for graduate-level instruction in dynamical systems and mathematical biology.

This paper is organized as follows. In Section 2, I derive the model. In Section 3, I establish a positively invariant (and absorbing) region and derive basic boundedness properties. Section 4 addresses the existence of equilibria, while Section 5 is devoted to their stability analysis. Bifurcations of the interior equilibrium are studied in Section 6. In Section 7, I discuss the implications of eco-evolutionary feedback. Section 8 contains the conclusions, and Section 9 outlines several directions for future research.

2. Model Formulation and Preliminaries

I consider discrete-time models as a natural framework for the study of populations with non-overlapping generations and evolutionary change occurring across successive reproductive cycles. In this context, nonlinear difference equations are widely used to describe density-dependent regulation, selection mechanisms, and feedbacks between ecological and evolutionary processes.

Among classical discrete-time population models, logistic-type equations occupy a central role due to their bounded state space and their ability to capture essential features of population growth under resource limitation. These models have also become canonical examples in the theory of nonlinear discrete dynamical systems, serving as a baseline for the analysis of more complex coupled dynamics.

I recall that the discrete logistic model for a single species can be written as

$$x(t+1)=bx\left(t\right)\left(1-cx\left(t\right)\right):=x\left(t\right)r\left(x\left(t\right)\right),$$

(1)

where $x\left(t\right)\ge 0$ denotes the population density (or abundance) at time unit t and $r\left(x\right)=b(1-cx)$ is the density-dependent per capita growth factor. The parameter $b>0$ represents the intrinsic growth rate at low density (since $r\left(0\right)=b$), whereas $c>0$ measures the strength of density dependence (crowding). In particular, the positive equilibrium of (1), when it exists, is

$$x^{*}={\displaystyle \frac{b-1}{bc}},$$

so larger c corresponds to a smaller characteristic population scale. Throughout, it is assumed that $c\in (0,1)$ and $0<b\le 4c$ and $b(1-c)\le 1$, under which the interval $[0,1]$ is forward invariant for (1). The dynamics of (1) are well documented in standard references on discrete dynamical systems; see, for instance, [18,19,20,21].

Motivated by eco-evolutionary models [3,4,6,14], I adopt the following Darwinian framework:

$$\left\{\begin{array}{c}x(t+1)=r\left(x\left(t\right),v,u\left(t\right)\right){|}_{v=u\left(t\right)}x\left(t\right),\hfill \\ u(t+1)=u\left(t\right)+{\sigma }^2{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }v}}ln\left(r\left(x\right(t),v,u(t\left)\right)\right){|}_{v=u\left(t\right)}.\hfill \end{array}\right.$$

Here, $ln\left(r\right(x,v,u\left)\right)$ is the fitness function and $r(x,v,u)$ is the growth rate contribution of an individual with trait value v in a population of density x and mean trait u. The parameter ${\sigma }^2>0$ represents the (additive) genetic variance and sets the timescale of evolutionary change.

In this paper I focus on the evolutionary logistic-type case in which the growth rate contribution takes the form

$$r(x,v,u)=b\left(v\right)\left(1-c(v-u)x\right),$$

where v is the inherited trait of a focal individual and u is the population mean trait. The trait-dependent density-free birth rate is modeled by a Gaussian-shaped function

$$b\left(v\right)=b_{0}exp\left(-{\displaystyle \frac{v^2}{2}}\right),$$

so that individuals with trait values closer to $v=0$ have higher baseline reproductive output. Since no reference scale is imposed on the phenotypic trait, the variance in $b\left(v\right)$ is set to 1 without loss of generality; with this normalization, $b\left(v\right)$ attains its unique maximum at $v=0$.

The coefficient c in the density-dependent term of r (motivated by logistic regulation) quantifies the per capita effect of intraspecific competition, that is, the reduction in an individual’s reproductive output caused by a single conspecific competitor. This competitive effect is assumed to depend on the trait difference $v-u$ and is taken in the special form

$$c=c(v-u),$$

where $c(·)$ is a positive, continuously differentiable function on $\mathbb{R}$. In particular, set

$$c_0:=c\left(0\right)>0,c_1:=c^{\prime }\left(0\right).$$

With these modeling choices, I obtain the fitness gradient

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }v}}ln\left(b\left(v\right)\left(1-c(v-u)x\right)\right).$$

Evaluating at $v=u$ yields

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }v}}\left(lnb\left(v\right)+ln\left(1-c(v-u)x\right)\right)=-v+{\displaystyle \frac{c^{\prime }(v-u)x}{1-c(v-u)x}},$$

and therefore

$${\left.{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }v}}ln\left(b\left(v\right)\left(1-c(v-u)x\right)\right)\right|}_{v=u}=-u+{\displaystyle \frac{c^{\prime }\left(0\right)x}{1-c\left(0\right)x}}.$$

With $b\left(v\right)=b_0{e}^{-v^2/2}$ and $c(v-u)$ satisfying $c_0:=c\left(0\right)>0$ and $c_1:=c^{\prime }\left(0\right)$, the resulting eco-evolutionary logistic-type model is

$$\begin{array}{cc}\hfill x(t+1)& =b_{0}x\left(t\right)\left(1-c_{0}x\left(t\right)\right)exp\left(-\frac{1}{2}u{\left(t\right)}^2\right),\hfill \\ \hfill u(t+1)& =(1-{\sigma }^2)u\left(t\right)+c_1{\sigma }^2{\displaystyle \frac{x\left(t\right)}{1-c_{0}x\left(t\right)}},\hfill \end{array}$$

(2)

where $t\in \mathbb{N}$ denotes discrete time. In particular, the update for $u\left(t\right)$ is well defined whenever $1-c_{0}x\left(t\right)>0$.

The variable $x\left(t\right)$ denotes the (normalized) population density at time t, and $u\left(t\right)\in \mathbb{R}$ is the mean value of a quantitative trait. The parameter $b_0>0$ is the baseline (density-free) reproductive rate, and $(1-c_{0}x\left(t\right))$ models logistic-type density regulation with crowding intensity $c_0>0$. Stabilizing selection around the optimal trait value $u=0$ is represented by the factor $exp\left(-\frac{1}{2}u{\left(t\right)}^2\right)$. The parameter ${\sigma }^2>0$ represents the (additive) genetic variance of the trait. In discrete time it plays the role of a step size in the mean trait update; for the analysis, it is convenient to restrict to $0<{\sigma }^2<2$, which keeps the linear part of the trait update, $(1-{\sigma }^2)u\left(t\right)$, within the standard nondegenerate range $-1<1-{\sigma }^2<1$. The parameter $c_1\in \mathbb{R}$ measures the strength (and direction) of eco-evolutionary feedback in the trait update.

Therefore, throughout this paper I assume

$$0<c_0<1,0<b_0\le 4c_0,b(1-c_0)\le 1,c_1\in \mathbb{R},0<{\sigma }^2<2,$$

(3)

and consider initial conditions $\left(x\right(0),u(0\left)\right)\in [0,1]\times \mathbb{R}$.

Under these assumptions, system (2) defines a nonlinear two-dimensional discrete-time dynamical system generated by the map

$$F:[0,1]\times \mathbb{R}\to [0,1]\times \mathbb{R},F(x,u)=\left(b_{0}x(1-c_{0}x)e^{-u^2/2},(1-{\sigma }^2)u+c_1{\sigma }^2{\displaystyle \frac{x}{1-c_{0}x}}\right).$$

In Section 3 I show that $x\left(t\right)\in [0,1]$ for all t whenever $x\left(0\right)\in [0,1]$ and that the full dynamics admit a compact absorbing positively invariant set. For an initial condition $\left(x\right(0),u(0\left)\right)\in [0,1]\times \mathbb{R}$, the trajectory is $(x\left(t\right),u\left(t\right))=F^t(x\left(0\right),u\left(0\right))$, $t\in \mathbb{N}$.

3. Invariant Region

To analyze the long-term dynamics of system (2), I first identify a compact positively invariant set in which all trajectories evolve. This provides a natural phase space for the subsequent analysis.

3.1. Positivity and Boundedness of the Population Variable

Given $x\left(0\right)\ge 0$ and $u\left(0\right)\in \mathbb{R}$, I observe that the population component remains non-negative for all $t\ge 0$. Indeed, since $b_0>0$ and $exp(-\frac{1}{2}u{\left(t\right)}^2)>0$, it follows from the first equation in (2) that $x(t+1)\ge 0$ whenever $x\left(t\right)\ge 0$.

Moreover, if $x\left(t\right)\in [0,1]$, then

$$x(t+1)=b_{0}x\left(t\right)(1-c_{0}x\left(t\right))exp\left(-\frac{1}{2}u{\left(t\right)}^2\right)\le b_{0}x\left(t\right)(1-c_{0}x\left(t\right)),$$

since $exp(-\frac{1}{2}u{\left(t\right)}^2)\le 1$. For $0<b_0\le 4c_0$ with $0<c_0<1$ such that $b(1-c_0)\le 1$, the logistic map $y\mapsto b_{0}y(1-c_{0}y)$ maps $[0,1]$ into itself, and therefore $x(t+1)\in [0,1]$ whenever $x\left(t\right)\in [0,1]$. In particular, if $x\left(0\right)\in [0,1]$, then $x\left(t\right)\in [0,1]$ for all $t\in \mathbb{N}$.

3.2. Boundedness of the Evolutionary Variable

Assume that $x\left(t\right)\in [0,1]$ for all $t\ge 0$. Then the nonlinear feedback term satisfies

$$0\le {\displaystyle \frac{x\left(t\right)}{1-c_{0}x\left(t\right)}}\le {\displaystyle \frac{1}{1-c_0}},$$

since $c_0\in (0,1)$. Consequently, the second equation in (2) yields

$$\left|u(t+1)\right|\le |1-{\sigma }^2\left|\right|u\left(t\right)|+{\displaystyle \frac{|c_1|{\sigma }^2}{1-c_0}}.$$

If $0<{\sigma }^2<2$, then $a:=|1-{\sigma }^2| <1$. Hence,

$$\left|u(t+1)\right|\le a\left|u\left(t\right)\right|+b,b:={\displaystyle \frac{|c_1|{\sigma }^2}{1-c_0}}.$$

By standard iteration of linear difference inequalities (see, e.g., [19,24]), for any $t\ge t_0$ it follows that

$$\begin{array}{cc}\hfill \left|u\right(t\left)\right|& \le a^{t-t_0}\left|u\left(t_0\right)\right|+b\sum _{j=0}^{t-t_0-1}{a}^j\hfill \\ \hfill & =a^{t-t_0}\left|u\left(t_0\right)\right|+{\displaystyle \frac{b(1-a^{t-t_0})}{1-a}}\hfill \\ \hfill & \le a^{t-t_0}\left|u\left(t_0\right)\right|+{\displaystyle \frac{b}{1-a}}.\hfill \end{array}$$

In particular, there exists a constant $M>0$ such that $\left|u\right(t\left)\right|\le M$ for any initial condition $u\left(0\right)\in \mathbb{R}$ and for all sufficiently large t. Thus the interval $[-M,M]$ is absorbing for the evolutionary dynamics. Hence, I may choose, for instance,

$$M\ge {\displaystyle \frac{|c_1|{\sigma }^2}{(1-c_0)(1-|1-{\sigma }^2\left|\right)}}.$$

3.3. A Positively Invariant Set

Combining the previous estimates yields the following result.

Theorem 1.

Assume that

$$0<b_0\le 4c_0,c_0\in (0,1),b(1-c_0)\le 1,c_1\in \mathbb{R},0<{\sigma }^2<2.$$

Then the set

$$\mathcal{D}=[0,1]\times [-M,M],$$

with $M>0$ chosen as above, is compact, absorbing, and positively invariant under the dynamics of system (2).

Proof. 

Let $\left(x\right(t),u(t\left)\right)\in \mathcal{D}$. From the first equation in (2) and the bounds derived above, it follows that $x(t+1)\in [0,1]$. Similarly, the estimate on $u(t+1)$ ensures that $u(t+1)\in [-M,M]$ whenever $u\left(t\right)\in [-M,M]$. Therefore, $\left(x\right(t+1),u(t+1\left)\right)\in \mathcal{D}$, which proves the claim. □

The set $\mathcal{D}$ is a compact and biologically meaningful phase space for (2), and it confines all trajectories for the subsequent analysis.

3.4. Global Attractor

Now, I establish the existence of a global attractor for system (2).

Theorem 2.

Assume

$$0<b_0\le 4c_0,c_0\in (0,1),b(1-c_0)\le 1,c_1\in \mathbb{R},0<{\sigma }^2<2.$$

Then system (2) generates a continuous discrete-time dynamical system on $[0,1]\times \mathbb{R}$ possessing a compact global attractor $\mathcal{A}\subset [0,1]\times \mathbb{R}$.

Proof. 

From Theorem 1, there exists a compact set $\mathcal{D}=[0,1]\times [-M,M]$, which is absorbing and positively invariant. Define

$$\mathcal{A}=\bigcap _{t=0}^{\infty }{F}^t\left(\mathcal{D}\right).$$

Since F is continuous and $F\left(\mathcal{D}\right)\subset \mathcal{D}$, the set $\mathcal{A}$ is nonempty, compact, and strictly invariant. Moreover, for any bounded set $B\subset [0,1]\times \mathbb{R}$, there exists $T\left(B\right)\in \mathbb{N}$ such that $F^t\left(B\right)\subset \mathcal{D}$ for all $t\ge T\left(B\right)$, implying that $\mathcal{A}$ attracts B. Hence $\mathcal{A}$ is a compact global attractor. □

4. Existence of Equilibria

In this section I characterize the fixed points of system (2) and provide conditions for the existence of a nontrivial equilibrium.

An equilibrium $(x,u)$ of system (2) satisfies

$$\begin{array}{cc}\hfill x& =b_{0}x(1-c_{0}x)exp\left(-\frac{1}{2}{u}^2\right),\hfill \\ \hfill u& =(1-{\sigma }^2)u+c_1{\sigma }^2{\displaystyle \frac{x}{1-c_{0}x}}.\hfill \end{array}$$

(4)

Theorem 3

(Equilibria). Assume

$$0<b_0\le 4c_0,c_0\in (0,1),b(1-c_0)\le 1,c_1\in \mathbb{R},0<{\sigma }^2<2.$$

Then the following statements hold.

1. 

System (2) admits the equilibrium $E_0=(0,0)$.

2. 

If $b_0\le 1$, then $E_0$ is the only equilibrium with $x\in [0,1]$. If $b_0>1$, then system (2) admits exactly two equilibria in $[0,1]\times \mathbb{R}$: the trivial equilibrium $E_0=(0,0)$ and a unique nontrivial equilibrium $E^{*}=(x^{*},u^{*})$ with $x^{*}\in (0,1]$, where

$$b_0(1-c_0{x}^{*})exp\left(-\frac{1}{2}{\left(u^{*}\right)}^2\right)=1,u^{*}=c_1{\displaystyle \frac{x^{*}}{1-c_0{x}^{*}}}.$$

Moreover, the sign of the evolutionary component $u^{*}$ is determined by the sign of $c_1$. In particular, if $c_1>0$ then $u^{*}>0$, if $c_1<0$ then $u^{*}<0$, and if $c_1=0$ then $u^{*}=0$.

Proof. 

It is immediate that $(x,u)=(0,0)$ satisfies the equilibrium conditions; hence $E_0=(0,0)$ is an equilibrium of system (2).

Assume now that $x>0$. Dividing the first equilibrium equation by x yields

$$b_0(1-c_{0}x)exp\left(-\frac{1}{2}{u}^2\right)=1.$$

(5)

The second equilibrium equation implies

$$u=c_1{\displaystyle \frac{x}{1-c_{0}x}}.$$

(6)

Substituting (6) into (5), one obtains the scalar equation

$$b_0(1-c_{0}x)exp\left(-\frac{1}{2}{c}_1^2{\displaystyle \frac{x^2}{{(1-c_{0}x)}^2}}\right)=1,x\in (0,1].$$

(7)

Define the continuous function on $[0,1]$,

$$\Phi \left(x\right)=b_0(1-c_{0}x)exp\left(-\frac{1}{2}{c}_1^2{\displaystyle \frac{x^2}{{(1-c_{0}x)}^2}}\right),x\in [0,1].$$

A direct computation gives

$${\Phi }^{\prime }\left(x\right)=-b_0\left(c_0+{\displaystyle \frac{c_1^{2}x}{{(1-c_{0}x)}^2}}\right)exp\left(-\frac{1}{2}{c}_1^2{\displaystyle \frac{x^2}{{(1-c_{0}x)}^2}}\right),$$

so ${\Phi }^{\prime }\left(x\right)<0$ for all $x\in [0,1]$ and therefore $\Phi$ is strictly decreasing on $[0,1]$. Moreover,

$$\Phi \left(0\right)=b_0>1,\underset{x\to 1}{lim}\Phi \left(x\right)=b_0(1-c_0)exp\left(-\frac{1}{2}{\displaystyle \frac{c_1^2}{{(1-c_0)}^2}}\right)<1.$$

Hence, by continuity and strict monotonicity, Equation (7) admits a unique solution $x^{*}\in (0,1]$. The corresponding value $u^{*}=c_1{x}^{*}/(1-c_0{x}^{*})$ uniquely determines the nontrivial equilibrium $E^{*}$.

Finally, since $x^{*}>0$, the sign of $u^{*}$ is the sign of $c_1$. □

5. Stability

In this section I investigate the stability properties of the equilibria of system (2). First, I establish the global asymptotic stability of the trivial equilibrium when $b_0\le 1$. Then I analyze the local stability of the nontrivial equilibrium by means of the linearization principle. Throughout this section, stability is understood in the sense of discrete-time dynamical systems, that is, in terms of the location of the eigenvalues of the Jacobian matrix with respect to the unit circle.

For the associated map F, the Jacobian matrix at a point $(x,u)$ is given by

$$DF(x,u)=\left(\begin{array}{cc}{b}_0(1-2c_{0}x)exp\left(-\frac{1}{2}{u}^2\right)& -b_{0}x(1-c_{0}x)uexp\left(-\frac{1}{2}{u}^2\right)\\ {\displaystyle \frac{c_1{\sigma }^2}{{(1-c_{0}x)}^2}}& 1-{\sigma }^2\end{array}\right).$$

5.1. Stability of the Trivial Equilibrium

The trivial equilibrium $E_0=(0,0)$ always exists and corresponds to population extinction. Evaluating the Jacobian at $E_0$ yields

$$DF\left(E_0\right)=\left(\begin{array}{cc}{b}_0& 0\\ c_1{\sigma }^2& 1-{\sigma }^2\end{array}\right).$$

The eigenvalues are therefore

$${\lambda }_1=b_0,{\lambda }_2=1-{\sigma }^2.$$

Since $0<{\sigma }^2<2$, it follows that $|{\lambda }_2|<1$. Hence the local asymptotic stability (LAS) of $E_0$ is determined solely by ${\lambda }_1$.

Proposition 1

(LAS of the origin). The trivial equilibrium $E_0$ is locally asymptotically stable if $0<b_0<1$ and unstable if $b_0>1$. At $b_0=1$, a change of stability occurs.

This threshold coincides with the classical persistence condition for discrete-time population models.

I next consider the global asymptotic stability (GAS) of the origin.

Theorem 4

(GAS of the origin). Let $c_0\in (0,1)$, $c_1\in \mathbb{R}$ and $0<{\sigma }^2<2$. Assume that $b_0\le 1$ and that the initial conditions satisfy $x\left(0\right)\in [0,1]$ and $u\left(0\right)\in \mathbb{R}$. Then, the extinction equilibrium $E_0$ of model (2) is globally asymptotically stable on $[0,1]\times \mathbb{R}$. In particular,

$$\underset{t\to \infty }{lim}(x\left(t\right),u\left(t\right))=(0,0).$$

Proof. 

If $x\left(0\right)=0$, then $x\left(t\right)=0$ for all t, and the trait equation reduces to

$$u(t+1)=(1-{\sigma }^2)u\left(t\right),$$

so that $u\left(t\right)\to 0$ since $|1-{\sigma }^2|<1$.

Assume now that $0<x\left(0\right)\le 1$. Then $0<c_{0}x\left(t\right)\le c_0<1$ for all t, since every factor in the x-equation lies in the unit interval. Thus, $0<1-c_{0}x\left(t\right)<1$ for all t. Moreover, since $exp(-\frac{1}{2}u{\left(t\right)}^2)\le 1$, I obtain for all $t\ge 0$

$$0<x(t+1)\le b_{0}x\left(t\right)(1-c_{0}x\left(t\right))\le b_{0}x\left(t\right).$$

(8)

If $b_0<1$, then iterating (8) yields $x\left(t\right)\le b_0^{t}x\left(0\right)$; hence $x\left(t\right)\to 0$. If $b_0=1$, then (8) yields

$$0<x(t+1)\le x\left(t\right)(1-c_{0}x\left(t\right))<x\left(t\right),$$

so $\left\{x\right(t\left)\right\}$ is strictly decreasing and bounded below by 0. Hence $x\left(t\right)\to L\ge 0$. Passing to the limit in the inequality $x(t+1)\le x\left(t\right)(1-c_{0}x\left(t\right))$ gives

$$L\le L(1-c_{0}L).$$

If $L>0$, division by L yields $1\le 1-c_{0}L$, a contradiction. Thus $L=0$.

It remains to show that $u\left(t\right)\to 0$ as $t\to \infty$. From the second equation in (2),

$$u(t+1)=(1-{\sigma }^2)u\left(t\right)+{\sigma }^2{c}_{1}f\left(t\right),f\left(t\right)={\displaystyle \frac{x\left(t\right)}{1-c_{0}x\left(t\right)}}.$$

Since $x\left(t\right)\to 0$ and f is continuous at 0, we have $f\left(t\right)\to 0$.

Iterating the recurrence [19,24] yields, for all $t\ge 1$,

$$u\left(t\right)={(1-{\sigma }^2)}^{t}u\left(0\right)+{\sigma }^2{c}_1\sum _{k=0}^{t-1}{(1-{\sigma }^2)}^{t-1-k}f\left(k\right).$$

(9)

Since $|1-{\sigma }^2|<1$, the first term in (9) converges to 0. For the convolution term, let $\rho :=|1-{\sigma }^2|<1$. Since $x\left(t\right)\in [0,1]$,

$$0\le f\left(t\right)={\displaystyle \frac{x\left(t\right)}{1-c_{0}x\left(t\right)}}\le {\displaystyle \frac{1}{1-c_0}}=:M_f,\mathrm{and}f\left(t\right)\to 0.$$

Fix $\epsilon >0$ and choose N such that $\left|f\right(k\left)\right|\le \epsilon$ for all $k\ge N$. For $t>N$, set

$$S_t:=\sum _{k=0}^{t-1}{\rho }^{t-1-k}\left|f\left(k\right)\right|.$$

Then

$$\begin{array}{cc}\hfill S_t& \le \sum _{k=0}^{N-1}{\rho }^{t-1-k}{M}_f+\sum _{k=N}^{t-1}{\rho }^{t-1-k}\epsilon \hfill \\ \hfill & =M_f\sum _{k=0}^{N-1}{\rho }^{t-1-k}+\epsilon \sum _{k=N}^{t-1}{\rho }^{t-1-k}\hfill \\ \hfill & =M_f{\rho }^{t-N}\sum _{k=0}^{N-1}{\rho }^{N-1-k}+\epsilon \sum _{k=N}^{t-1}{\rho }^{t-1-k}\hfill \\ \hfill & =M_f{\rho }^{t-N}\sum _{j=0}^{N-1}{\rho }^j+\epsilon \sum _{j=0}^{t-1-N}{\rho }^j\hfill \\ \hfill & \le M_f{\rho }^{t-N}\sum _{j=0}^{N-1}{\rho }^j+\epsilon \sum _{j=0}^{\infty }{\rho }^j\hfill \\ \hfill & =M_f{\rho }^{t-N}\sum _{j=0}^{N-1}{\rho }^j+{\displaystyle \frac{\epsilon }{1-\rho }}.\hfill \end{array}$$

where I used the reindexings $j=N-1-k$ in the first finite sum and $j=t-1-k$ in the second sum. Letting $t\to \infty$ yields ${lim\; sup}_{t\to \infty }{S}_t\le \epsilon /(1-\rho )$. Since $\epsilon >0$ is arbitrary, it follows that $S_t\to 0$.

Returning to (9), we obtain

$$\left|u\left(t\right)\right|\le {\rho }^t\left|u\left(0\right)\right|+|{\sigma }^2{c}_1|S_t\underset{t\to \infty }{\to }0,$$

and hence $u\left(t\right)\to 0$. □

Corollary 1.

Assume that $b_0\le 1$, $c_0\in (0,1)$, $c_1\in \mathbb{R}$ and $0<{\sigma }^2<2$. Then the global attractor $\mathcal{A}$ of system (2) reduces to the singleton

$$\mathcal{A}=\left\{\right(0,0\left)\right\}.$$

Proof. 

By Theorem 4, the equilibrium $(0,0)$ is globally asymptotically stable in $[0,1]\times \mathbb{R}$; that is,

$$\underset{t\to \infty }{lim}{F}^t(x,u)=(0,0)\mathrm{for}\mathrm{all}(x,u)\in [0,1]\times \mathbb{R}.$$

Since $\mathcal{A}$ is compact, invariant and attracts all bounded sets, it must be contained in the basin of attraction of $(0,0)$. Moreover, invariance implies that $\mathcal{A}$ cannot contain any point other than $(0,0)$. Hence $\mathcal{A}=\left\{\right(0,0\left)\right\}$. □

5.2. Local Stability of the Nontrivial Equilibrium

Assume that $b_0>1$ so that system (2) admits a unique nontrivial equilibrium

$$E^{*}=(x^{*},u^{*}),x^{*}\in (0,1],u^{*}={\displaystyle \frac{c_1{x}^{*}}{1-c_0{x}^{*}}}.$$

The equilibrium values satisfy

$$b_0(1-c_0{x}^{*})exp\left(-\frac{1}{2}{\left(u^{*}\right)}^2\right)=1.$$

(10)

Linearizing system (2) at $E^{*}$ and using (10), the Jacobian matrix takes the form

$$DF\left(E^{*}\right)=\left(\begin{array}{cc}{\displaystyle \frac{1-2c_0{x}^{*}}{1-c_0{x}^{*}}}& -x^{*}{u}^{*}\\ {\displaystyle \frac{c_1{\sigma }^2}{{(1-c_0{x}^{*})}^2}}& 1-{\sigma }^2\end{array}\right).$$

Let $\tau$ and $\Delta$ denote the trace and determinant of $DF\left(E^{*}\right)$, respectively. A direct computation yields

$$\tau ={\displaystyle \frac{1-2c_0{x}^{*}}{1-c_0{x}^{*}}}+(1-{\sigma }^2),$$

(11)

and

$$\Delta ={\displaystyle \frac{1-2c_0{x}^{*}}{1-c_0{x}^{*}}}(1-{\sigma }^2)+{\displaystyle \frac{c_1^2{\sigma }^2{\left(x^{*}\right)}^2}{{(1-c_0{x}^{*})}^3}},$$

(12)

where we have used the identity $u^{*}=c_1{x}^{*}/(1-c_0{x}^{*})$.

The local asymptotic stability of $E^{*}$ is determined by the Jury [25] conditions for two-dimensional discrete-time systems, namely

$$\left(\mathrm{J}1\right):1+\tau +\Delta >0,\left(\mathrm{J}2\right):1-\tau +\Delta >0,\left(\mathrm{J}3\right):1-\Delta >0.$$

(13)

Substituting (11) and (12) into (13) and simplifying, we obtain the following result.

Proposition 2.

The nontrivial equilibrium $E^{*}$ is locally asymptotically stable if and only if the following conditions are satisfied:

$$\begin{array}{cc}\hfill \left(\mathrm{J}1\right)& {\displaystyle \frac{(2-3c_0{x}^{*})(2-{\sigma }^2)}{1-c_0{x}^{*}}}+{\displaystyle \frac{c_1^2{\sigma }^2{\left(x^{*}\right)}^2}{{(1-c_0{x}^{*})}^3}}>0,\hfill \\ \hfill \left(\mathrm{J}3\right)& {\displaystyle \frac{{\sigma }^2+(1-2{\sigma }^2)c_0{x}^{*}}{1-c_0{x}^{*}}}-{\displaystyle \frac{c_1^2{\sigma }^2{\left(x^{*}\right)}^2}{{(1-c_0{x}^{*})}^3}}>0.\hfill \end{array}$$

Proof. 

It is a straightforward computation to obtain conditions (J1) and (J3) by substituting (11) and (12) into (13) and simplifying.

Moreover, condition (J2) in (13) is equivalent to

$${\displaystyle \frac{c_0{\sigma }^2{x}^{*}}{1-c_0{x}^{*}}}+{\displaystyle \frac{c_1^2{\sigma }^2{\left(x^{*}\right)}^2}{{(1-c_0{x}^{*})}^3}}>0.$$

Since $0<c_0<1$, $0<{\sigma }^2<2$, and $x^{*}\in (0,1]$ imply $1-c_0{x}^{*}>0$, both terms are non-negative and at least one is strictly positive whenever ${\sigma }^2>0$ and $x^{*}>0$. Hence (J2) holds automatically for all admissible parameter values, and the local asymptotic stability of $E^{*}$ is characterized by (J1) and (J3) alone. □

Remark 1.

The stability of the nontrivial equilibrium $E^{*}$ is determined by the interplay between density dependence, genetic variance, and the strength of eco-evolutionary feedback. In particular, increasing ${\sigma }^2$ reduces damping in the evolutionary component, while increasing $|c_1|$ amplifies nonlinear feedback effects. Loss of stability may occur through a flip (period-doubling) bifurcation when condition (J1) fails or through a Neimark–Sacker bifurcation when condition (J3) is violated. These mechanisms illustrate how evolutionary feedback can destabilize otherwise stable ecological dynamics.

6. Bifurcations

In this section I investigate the bifurcations of the interior fixed point of system (2). The intrinsic growth rate $b_0$ plays a fundamental role: it governs the extinction–persistence transition and the appearance of a nontrivial equilibrium. Thus $b_0$ is taken as the primary bifurcation parameter. Once persistence is established, the genetic variance ${\sigma }^2$ and the eco-evolutionary feedback strength $c_1$ act as secondary parameters that can destabilize the interior equilibrium and generate more complex dynamics.

For a smooth two-dimensional map depending on a parameter, a fixed point may lose stability in three basic ways. A fold (saddle-node) bifurcation is associated with a real eigenvalue crossing $+1$; equivalently

$$1-\tau +\Delta =0,\mathrm{i}.\mathrm{e}.,\Delta =\tau -1,$$

where $\tau =\mathrm{tr}DF(x^{*},u^{*})$ and $\Delta =detDF(x^{*},u^{*})$.

A flip (period-doubling) bifurcation occurs when a real eigenvalue crosses $-1$; equivalently

$$1+\tau +\Delta =0,\mathrm{i}.\mathrm{e}.,\Delta =-\tau -1.$$

Finally, a Neimark–Sacker bifurcation occurs when a complex-conjugate pair of eigenvalues crosses the unit circle, which (for a two-dimensional map) is signaled by

$$\Delta =1\mathrm{and}-2<\tau <2.$$

In the present model, the loss of stability at $b_0=1$ is not a generic fold: rather, $E_0=(0,0)$ lies on a branch of equilibria (the extinction equilibrium), and the bifurcation is of transcritical type under appropriate nondegeneracy conditions. This is analyzed next.

6.1. Transcritical Bifurcation

I now classify the bifurcation that occurs in system (2) as $b_0$ passes through the critical value $b_0=1$.

Theorem 5

(Transcritical bifurcation). Let $c_0\in (0,1)$, $c_1\in \mathbb{R}$, and $0<{\sigma }^2<2$. Consider system (2) and the trivial equilibrium $E_0=(0,0)$. Then, at $b_0=1$ the Jacobian at $E_0$ has eigenvalues

$${\lambda }_1=1,{\lambda }_2=1-{\sigma }^2\in (-1,1).$$

Moreover, for $b_0>1$ sufficiently close to 1, there exists a nontrivial equilibrium $E^{*}=(x^{*},u^{*})$ with $x^{*}>0$ that emerges continuously from $E_0$ and satisfies

$$u^{*}={\displaystyle \frac{c_1{x}^{*}}{1-c_0{x}^{*}}}.$$

In particular, writing $b_0=1+\epsilon$ with $\epsilon >0$ small, one has the expansions

$$x^{*}={\displaystyle \frac{\epsilon }{c_0}}+\mathcal{O}\left({\epsilon }^2\right),u^{*}={\displaystyle \frac{c_1}{c_0}}\epsilon +\mathcal{O}\left({\epsilon }^2\right).$$

Proof. 

The Jacobian at $E_0$ is

$$DF\left(E_0\right)=\left(\begin{array}{cc}{b}_0& 0\\ c_1{\sigma }^2& 1-{\sigma }^2\end{array}\right),$$

with eigenvalues ${\lambda }_1=b_0$ and ${\lambda }_2=1-{\sigma }^2$. Hence at $b_0=1$, ${\lambda }_1=1$ and ${\lambda }_2\in (-1,1)$.

For $b_0>1$, equilibria with $x>0$ satisfy (5)–(6); equivalently

$$b_0(1-c_{0}x)exp\left(-\frac{1}{2}{u}^2\right)=1,u={\displaystyle \frac{c_{1}x}{1-c_{0}x}}.$$

Let $b_0=1+\epsilon$ with $\epsilon >0$ small. Seeking a small equilibrium, set $x=\alpha \epsilon +\mathcal{O}\left({\epsilon }^2\right)$ and $u=\beta \epsilon +\mathcal{O}\left({\epsilon }^2\right)$. Using

$$exp\left(-\frac{1}{2}{u}^2\right)=1+\mathcal{O}\left({\epsilon }^2\right),$$

the first equilibrium condition becomes

$$(1+\epsilon )\left(1-c_{0}x\right)\left(1+\mathcal{O}\left({\epsilon }^2\right)\right)=1,$$

That is,

$$\epsilon -c_{0}x+\mathcal{O}\left({\epsilon }^2\right)=0⟹x={\displaystyle \frac{\epsilon }{c_0}}+\mathcal{O}\left({\epsilon }^2\right).$$

Substituting into $u={\displaystyle \frac{c_{1}x}{1-c_{0}x}}$ and using $1-c_{0}x=1+\mathcal{O}\left(\epsilon \right)$ gives

$$u={\displaystyle \frac{c_1}{c_0}}\epsilon +\mathcal{O}\left({\epsilon }^2\right).$$

Thus a nontrivial equilibrium branch emerges from $E_0$ for $b_0>1$ close to 1, and the transcritical nature follows from the exchange of stability between $E_0$ and $E^{*}$. □

Remark 2.

The threshold $b_0=1$ corresponds to the classical persistence condition in discrete-time population models: for $b_0\le 1$ the extinction equilibrium $E_0$ attracts all trajectories in the biologically relevant region, while for $b_0>1$ a nontrivial equilibrium branch emerges from $E_0$. Subsequent destabilization of $E^{*}$ may occur via flip (period-doubling) or Neimark–Sacker bifurcations, driven by ${\sigma }^2$ and $c_1$. Figure 1 illustrates the transcritical bifurcation for the parameter set used throughout the numerical examples.

6.2. Period-Doubling (Flip) Bifurcation

Assume that $b_0>1$ so that the interior equilibrium $E^{*}=(x^{*},u^{*})$ exists with $x^{*}\in (0,1]$ and

$$u^{*}={\displaystyle \frac{c_1{x}^{*}}{1-c_0{x}^{*}}}.$$

Recall that the trace and determinant of the Jacobian matrix $DF\left(E^{*}\right)$ are given by

$$\tau ={\displaystyle \frac{1-2c_0{x}^{*}}{1-c_0{x}^{*}}}+1-{\sigma }^2,$$

(14)

and

$$\Delta ={\displaystyle \frac{1-2c_0{x}^{*}}{1-c_0{x}^{*}}}(1-{\sigma }^2)+{\displaystyle \frac{c_1^2{\sigma }^2{\left(x^{*}\right)}^2}{{(1-c_0{x}^{*})}^3}},$$

(15)

A period-doubling (flip) bifurcation occurs when a real eigenvalue of $DF\left(E^{*}\right)$ crosses $-1$ as the parameter $b_0$ varies. In terms of $\tau$ and $\Delta$, this corresponds to the condition

$$1+\tau +\Delta =0,$$

(16)

together with $|\Delta |<1$, which ensures that the second eigenvalue remains strictly inside the unit circle.

Substituting (14) and (15) into (16), the flip condition reduces to the scalar equation

$${\displaystyle \frac{(2-3c_0{x}^{*})(2-{\sigma }^2)}{1-c_0{x}^{*}}}+{\displaystyle \frac{c_1^2{\sigma }^2{\left(x^{*}\right)}^2}{{(1-c_0{x}^{*})}^3}}=0.$$

(17)

Theorem 6

(Period-doubling bifurcation). Suppose that there exists $b_0=b_0^{*}>1$ such that the interior equilibrium $E^{*}$ exists and satisfies the flip condition (17) together with $|\Delta (b_0^{*}\left)\right|<1$. Assume, moreover, that the flip transversality condition holds; i.e.,

$${\left.{\displaystyle \frac{d}{d{b}_0}}\left(1+\tau \left(b_0\right)+\Delta \left(b_0\right)\right)\right|}_{b_0=b_0^{*}}\ne 0.$$

Then system (2) undergoes a period-doubling (flip) bifurcation at $b_0=b_0^{*}$.

Furthermore, let c denote the flip normal form coefficient given by (A4). If $c<0$ (respectively $c>0$), then the flip bifurcation is supercritical (respectively subcritical).

Proof. 

At $b_0=b_0^{*}$, the characteristic polynomial of $DF\left(E^{*}\right)$ reads

$${\lambda }^2-\tau \lambda +\Delta =0,$$

and condition (16) implies that ${\lambda }_1=-1$ is a simple eigenvalue, while the second eigenvalue satisfies $|{\lambda }_2| <1$.

To determine transversality, I differentiate the characteristic equation implicitly with respect to $b_0$ along the eigenvalue branch ${\lambda }_1\left(b_0\right)$, which yields

$$(2{\lambda }_1-\tau ){\lambda }_1^{\prime }={\tau }^{\prime }\left(b_0\right){\lambda }_1-{\Delta }^{\prime }\left(b_0\right).$$

Evaluating at $b_0=b_0^{*}$, where ${\lambda }_1=-1$ and $\tau =-1-\Delta$, we obtain

$$(\Delta -1){\lambda }_1^{\prime }=-\left({\Delta }^{\prime }+{\tau }^{\prime }\right).$$

Since ${\displaystyle \frac{d{x}^{*}}{d{b}_0}}\ne 0$ by the implicit function theorem, the equality ${\Delta }^{\prime }\left(b_0^{*}\right)+{\tau }^{\prime }\left(b_0^{*}\right)=0$ holds only if $P\left(x^{*}\right)=0$. As shown in Appendix A, P is a polynomial (of degree at most two); hence the condition $P\left(x^{*}\right)=0$ defines a nongeneric subset of the parameter space. Therefore the transversality condition ${\lambda }_1^{\prime }\left(b_0^{*}\right)\ne 0$ holds for generic parameter values.

Now I turn to the normal form and criticality. Reducing the dynamics to the one-dimensional critical center manifold associated with the eigenvalue ${\lambda }_1=-1$ (see [23]) yields the flip normal form

$$z_{n+1}=-z_n+c{z}_n^3+\mathcal{O}\left(z_n^4\right),$$

where c is the flip normal form coefficient. Following Kuznetsov, c can be computed by the invariant formula (A4); see Appendix A for the explicit expressions of A, B, C, p, and q in the present model.

The sign of c determines the criticality: if $c<0$, the flip bifurcation is supercritical and a locally asymptotically stable period-two orbit is born at $b_0=b_0^{*}$; if $c>0$, the flip bifurcation is subcritical and the emerging period-two orbit is unstable. □

Example 1.

Consider system (2) with

$$c_0=0.85,c_1=0.15,{\sigma }^2=0.6,$$

and take $b_0$ as the bifurcation parameter. Solving the equilibrium equations together with the flip condition $1+\tau +\Delta =0$ yields

$$b_0^{*}\approx 3.46231,E^{*}=(x^{*},u^{*})\approx (0.809853,0.389822).$$

At $b_0=b_0^{*}$ the Jacobian at $E^{*}$ satisfies

$$\tau \left(b_0^{*}\right)\approx -0.808989,\Delta \left(b_0^{*}\right)\approx -0.191011,1+\tau \left(b_0^{*}\right)+\Delta \left(b_0^{*}\right)\approx 0,$$

and $|\Delta (b_0^{*}\left)\right|<1$, so one eigenvalue crosses $-1$ while the second remains strictly inside the unit circle. The flip normal form coefficient computed from (A4) is

$$c\approx 8.41187>0,$$

so Theorem 6 predicts a subcritical flip bifurcation.

Moreover, a numerical finite-difference approximation yields

$${\left.{\displaystyle \frac{d}{d{b}_0}}\left(1+\tau \left(b_0\right)+\Delta \left(b_0\right)\right)\right|}_{b_0=b_0^{*}}\approx -0.792577\ne 0,$$

confirming the transversality condition in Theorem 6.

Consistently, for $b_0=b_0^{*}+{10}^{-3}$ the period-two orbit born at the flip is numerically unstable: solving $F^2\left(z\right)=z$ near $E^{*}$ and evaluating the monodromy matrix $D\left(F^2\right)\left(z\right)=DF\left(F\left(z\right)\right)DF\left(z\right)$ yields $max\left|\mu \right|\approx 1.00133>1$. Figure 2 provides a numerical illustration.

6.3. Neimark–Sacker Bifurcation

In this subsection I study the occurrence of a Neimark–Sacker bifurcation in system (2). The analysis follows the standard local theory for maps, as presented in [23]. For recent studies of concrete discrete-time models exhibiting Neimark–Sacker (discrete Hopf) bifurcations, I refer, for instance, to discrete-time epidemic models [26,27], predator–prey and related ecological maps [28,29,30,31], discrete neural systems [32], and coupled-map constructions [33].

Assume that $b_0>1$ so that the interior equilibrium $E^{*}=(x^{*},u^{*})$ exists with $x^{*}\in (0,1]$ and

$$u^{*}={\displaystyle \frac{c_1{x}^{*}}{1-c_0{x}^{*}}}.$$

A Neimark–Sacker bifurcation (discrete Hopf bifurcation) occurs when a complex-conjugate pair of eigenvalues of the Jacobian matrix $DF\left(E^{*}\right)$ crosses the unit circle with nonzero speed as a parameter varies.

Recall that the Jacobian at $E^{*}$ is

$$DF\left(E^{*}\right)=\left(\begin{array}{cc}{\displaystyle \frac{1-2c_0{x}^{*}}{1-c_0{x}^{*}}}& -x^{*}{u}^{*}\\ {\displaystyle \frac{c_1{\sigma }^2}{{(1-c_0{x}^{*})}^2}}& 1-{\sigma }^2\end{array}\right),u^{*}={\displaystyle \frac{c_1{x}^{*}}{1-c_0{x}^{*}}}.$$

Let $\tau \left(b_0\right)=\mathrm{tr}DF\left(E^{*}\right)$ and $\Delta \left(b_0\right)=detDF\left(E^{*}\right)$. As computed in Section 5.2,

$$\tau ={\displaystyle \frac{1-2c_0{x}^{*}}{1-c_0{x}^{*}}}+(1-{\sigma }^2),$$

(18)

and

$$\Delta ={\displaystyle \frac{1-2c_0{x}^{*}}{1-c_0{x}^{*}}}(1-{\sigma }^2)+{\displaystyle \frac{c_1^2{\sigma }^2{\left(x^{*}\right)}^2}{{(1-c_0{x}^{*})}^3}}.$$

(19)

For a smooth planar map, a Neimark–Sacker bifurcation at $E^{*}$ is detected by the spectral conditions

$$\Delta \left({\overline{b}}_0\right)=1,-2<\tau \left({\overline{b}}_0\right)<2,$$

(20)

for some parameter value $b_0={\overline{b}}_0$. Under (20), the eigenvalues satisfy

$${\lambda }_{1,2}\left({\overline{b}}_0\right)=e^{\pm i\theta },\theta \in (0,\pi ),\tau \left({\overline{b}}_0\right)=2cos\theta .$$

Using (19), the Neimark–Sacker condition $\Delta \left({\overline{b}}_0\right)=1$ is equivalent to

$${\displaystyle \frac{{\sigma }^2+(1-2{\sigma }^2)c_0{x}^{*}}{1-c_0{x}^{*}}}={\displaystyle \frac{c_1^2{\sigma }^2{\left(x^{*}\right)}^2}{{(1-c_0{x}^{*})}^3}},$$

(21)

together with $-2<\tau \left({\overline{b}}_0\right)<2$.

Theorem 7

(Neimark–Sacker bifurcation). Assume that $b_0>1$ so that the interior equilibrium $E^{*}$ exists. Suppose that there exists $b_0={\overline{b}}_0$ such that

$$\Delta \left({\overline{b}}_0\right)=1,-2<\tau \left({\overline{b}}_0\right)<2.$$

Assume in addition that:

1. 

(Transversality) ${\Delta }^{\prime }\left({\overline{b}}_0\right)\ne 0$;

2. 

(Non-resonance) $\tau \left({\overline{b}}_0\right)\notin \{-1,0,1\}$ (equivalently, $\theta \notin \{\pi /3,\pi /2,2\pi /3\}$);

3. 

(Nondegeneracy) the first Lyapunov coefficient ${\mathcal{l}}_1$ at $b_0={\overline{b}}_0$ satisfies ${\mathcal{l}}_1\ne 0$.

Then system (2) undergoes a Neimark–Sacker bifurcation of $E^{*}$ at $b_0={\overline{b}}_0$. Moreover, the bifurcation is supercritical (respectively, subcritical) if ${\mathcal{l}}_1<0$ (respectively, ${\mathcal{l}}_1>0$), producing a stable (respectively, unstable) invariant closed curve for $b_0$ on the side where $|{\lambda }_{1,2}| >1$ (respectively, $|{\lambda }_{1,2}| <1$).

I now express the transversality condition in a convenient explicit form.

Theorem 8

(Transversality criterion). Assume (20) holds at $b_0={\overline{b}}_0$ and that $0<{\sigma }^2<2$, $c_0\in (0,1)$, and $c_1\ne 0$. Then

$${\left.{\displaystyle \frac{d}{d{b}_0}}\left|{\lambda }_{1,2}\left(b_0\right)\right|\right|}_{b_0={\overline{b}}_0}\ne 0\mathit{if}\mathit{and}\mathit{only}\mathit{if}{\Delta }^{\prime }\left({\overline{b}}_0\right)\ne 0.$$

Moreover,

$${\Delta }^{\prime }\left(b_0\right)={\displaystyle \frac{d{x}^{*}}{d{b}_0}}\left[-(1-{\sigma }^2){\displaystyle \frac{c_0}{{(1-c_0{x}^{*})}^2}}+{\displaystyle \frac{c_1^2{\sigma }^2{x}^{*}(2+c_0{x}^{*})}{{(1-c_0{x}^{*})}^4}}\right],$$

(22)

and therefore transversality holds whenever

$$P_{NS}\left(x^{*}\right):=-(1-{\sigma }^2)c_0{(1-c_0{x}^{*})}^2+c_1^2{\sigma }^2{x}^{*}(2+c_0{x}^{*})\ne 0.$$

(23)

In particular, the degeneracy condition $P_{NS}\left(x^{*}\right)=0$ defines an algebraic subset of codimension one in the parameter space; hence transversality holds for generic parameter values.

Proof. 

At a Neimark–Sacker point the eigenvalues are complex conjugates and satisfy $|{\lambda }_{1,2}{|}^2=\Delta$. Differentiating gives

$$2\left|{\lambda }_{1,2}\right|{\displaystyle \frac{d}{d{b}_0}}\left|{\lambda }_{1,2}\right|={\Delta }^{\prime }\left(b_0\right),$$

and since $\Delta \left({\overline{b}}_0\right)=1$ we obtain

$${\left.{\displaystyle \frac{d}{d{b}_0}}\left|{\lambda }_{1,2}\right|\right|}_{b_0={\overline{b}}_0}={\displaystyle \frac{1}{2}}{\Delta }^{\prime }\left({\overline{b}}_0\right),$$

proving the equivalence.

Next, since $x^{*}=x^{*}\left(b_0\right)$ depends smoothly on $b_0$ along the interior branch, ${\Delta }^{\prime }\left(b_0\right)={\displaystyle \frac{d\Delta }{dx}}\left(x^{*}\right){\displaystyle \frac{d{x}^{*}}{d{b}_0}}$. Differentiating (19) with respect to x yields

$${\displaystyle \frac{d\Delta }{dx}}\left(x\right)=-(1-{\sigma }^2){\displaystyle \frac{c_0}{{(1-c_{0}x)}^2}}+{\displaystyle \frac{c_1^2{\sigma }^{2}x(2+c_{0}x)}{{(1-c_{0}x)}^4}}.$$

Substituting $x=x^{*}$ gives (22), and multiplying by the strictly positive factor ${(1-c_0{x}^{*})}^4$ yields the polynomial condition (23). The final statement follows since $P_{NS}\left(x^{*}\right)=0$ is a single algebraic equation. □

Next, I address the non-resonance condition.

Proposition 3

(Non-resonance condition). Assume (20). Then the only possible resonances of order $k\le 4$ correspond to $\tau \left({\overline{b}}_0\right)\in \{-1,0,1\}$. In particular, if $\tau \left({\overline{b}}_0\right)\notin \{-1,0,1\}$ then no resonance of order $k=1,2,3,4$ occurs.

Proof. 

With $\Delta \left({\overline{b}}_0\right)=1$, we have ${\lambda }_{1,2}=e^{\pm i\theta }$ and $\tau =2cos\theta$. Resonances ${\lambda }^k=1$ for $k\le 4$ correspond to $\theta \in \{0,\pi ,\pi /2,2\pi /3\}$, giving $\tau \in \{\pm 2,0,\pm 1\}$. The condition $-2<\tau <2$ excludes $\pm 2$, leaving $\{-1,0,1\}$. □

Next, I consider the nondegeneracy condition and the fundamental first Lyapunov coefficient. The nondegeneracy condition is determined by the first Lyapunov coefficient ${\mathcal{l}}_1$, which can be computed from second and third derivatives of the map at $E^{*}$ using standard normal form theory for planar maps [23]. We provide the explicit formula and the required derivatives for the present model in Appendix B. In applications below, ${\mathcal{l}}_1$ is evaluated numerically at the Neimark–Sacker point to determine the criticality (supercritical/subcritical) of the bifurcation.

Remark 3.

Condition (21) together with $-2<\tau \left({\overline{b}}_0\right)<2$ determines candidate Neimark–Sacker points. The transversality condition reduces to $P_{NS}\left(x^{*}\right)\ne 0$, and non-resonance is ensured by excluding the isolated values $\tau \left({\overline{b}}_0\right)\in \{-1,0,1\}$. Finally, ${\mathcal{l}}_1\ne 0$ holds for generic parameters (otherwise a codimension-one degeneracy occurs) and its sign determines whether a stable or unstable invariant closed curve is born.

Example 2.

Consider system (2) with

$$c_0=0.2,c_1=5,{\sigma }^2=1.1,$$

and take $b_0$ as the bifurcation parameter. Solving the fixed-point equations $F(x,u)=(x,u)$ yields the interior equilibrium branch $E^{*}\left(b_0\right)=(x^{*}\left(b_0\right),u^{*}\left(b_0\right))$.

Neimark–Sacker point. Solving simultaneously the equilibrium equations and the Neimark–Sacker condition $\Delta \left(b_0\right)=detDF\left(E^{*}\left(b_0\right)\right)=1$ yields

$${\overline{b}}_0\approx 1.67843,E^{*}\left({\overline{b}}_0\right)\approx (0.188463,0.979225).$$

At $b_0={\overline{b}}_0$ we obtain

$$\tau \left({\overline{b}}_0\right)\approx 0.860831\in (-2,2),\Delta \left({\overline{b}}_0\right)=1,\tau {\left({\overline{b}}_0\right)}^2-4\Delta \left({\overline{b}}_0\right)\approx -3.25897<0,$$

so the critical eigenvalues form a complex-conjugate pair on the unit circle,

$${\lambda }_{1,2}\approx 0.430416\pm 0.902631i=e^{\pm i\theta },\theta \approx 1.12584.$$

Transversality. A finite-difference approximation gives

$${\Delta }^{\prime }\left({\overline{b}}_0\right)\approx 1.33761\ne 0,$$

confirming transversal crossing.

Criticality. The first Lyapunov coefficient computed from the invariant formula (A19) (Appendix B) is

$${\mathcal{l}}_1\approx 1.9082\times {10}^{12}>0,$$

Hence the Neimark–Sacker bifurcation is subcritical. Consequently, the invariant closed curve born at the bifurcation is unstable, and nonlinear dynamics near ${\overline{b}}_0$ may display bistability and large-amplitude oscillations depending on initial conditions. Figure 3 provides a numerical illustration of the emergence of oscillatory population–trait dynamics as $b_0$ is increased past the Neimark–Sacker threshold.

7. Dynamical Consequences of Evolutionary Feedback

In this section, I highlight the qualitative impact of introducing evolutionary dynamics into a classical discrete-time logistic framework. In the absence of eco-evolutionary feedback ($c_1=0$), the trait dynamics decouple and satisfy $u(t+1)=(1-{\sigma }^2)u\left(t\right)$, so $u\left(t\right)\to 0$ for $0<{\sigma }^2<2$. Consequently, the asymptotic population dynamics are governed by the corresponding density-regulated one-dimensional map, whose behavior is well understood [20] in discrete time and includes transitions from fixed points to periodic and more complex regimes as $b_0$ increases.

Introducing evolutionary feedback ($c_1\ne 0$) fundamentally alters this picture. The coupling between population density and trait evolution, mediated by the feedback coefficient $c_1$ and the genetic variance ${\sigma }^2$, yields a genuinely two-dimensional map with richer dynamical behavior. While the threshold $b_0=1$ still marks the transition from extinction to persistence through a transcritical bifurcation, the stability of the interior equilibrium is no longer guaranteed.

The analysis shows that increasing either the strength of eco-evolutionary feedback or the genetic variance may destabilize the interior equilibrium. Loss of stability can occur through a flip (period-doubling) bifurcation, leading to the creation of stable or unstable two-cycles, or through a Neimark–Sacker bifurcation, associated with the appearance of invariant closed curves (stable or unstable) and quasiperiodic population–trait oscillations. In the supercritical case a stable invariant closed curve is born locally at the bifurcation, whereas in the subcritical case the local curve is unstable and multistability and large-amplitude oscillations may arise from nonlinear global effects. These mechanisms are absent in the purely ecological setting and arise from the interaction between ecological and evolutionary timescales.

From a biological perspective, these results indicate that rapid evolution can act as a destabilizing force, generating sustained oscillations or complex fluctuations even in environments where population dynamics alone might predict convergence to equilibrium. From a dynamical systems viewpoint, the model provides a minimal setting in which eco-evolutionary feedback induces classical discrete-time bifurcations and higher-dimensional attractors.

Taken together, these results demonstrate that incorporating evolution into population models is not merely a quantitative refinement, but can lead to qualitatively new dynamical regimes, emphasizing the central role of eco-evolutionary feedbacks in shaping long-term population behavior.

8. Conclusions

In this work I studied a discrete-time eco-evolutionary logistic-type model obtained by coupling density-regulated population growth with trait dynamics driven by selection and genetic variance. I provided a rigorous dynamical analysis of the model, including the existence of equilibria, global asymptotic stability of the extinction equilibrium, and a detailed classification of local bifurcations of the interior fixed point.

I showed that the extinction equilibrium is globally asymptotically stable when the intrinsic growth rate satisfies $b_0\le 1$, while for $b_0>1$ a nontrivial equilibrium emerges through a transcritical bifurcation. Once persistence is established, the stability of the interior equilibrium is strongly affected by eco-evolutionary feedback. Depending on the feedback strength and on the genetic variance, the equilibrium may lose stability through either a flip (period-doubling) bifurcation or a Neimark–Sacker bifurcation, leading to oscillatory regimes. In the Neimark–Sacker scenario, invariant closed curves and quasiperiodic dynamics may occur, with the detailed outcome depending on the criticality of the bifurcation and on nonlinear global effects.

These results highlight how even a minimal evolutionary component can generate qualitatively new dynamical regimes in discrete-time population models. From both mathematical and biological perspectives, the model illustrates how eco-evolutionary interactions may destabilize ecological equilibria and give rise to persistent fluctuations.

9. Future Directions

I outline several natural extensions of the present work that deserve further investigation.

A natural and challenging open problem concerns the global stability of the interior equilibrium. While the analysis presented in this paper provides sharp conditions for local asymptotic stability and a complete description of the local bifurcation structure, global asymptotic stability cannot be expected in general for discrete-time systems of this type. Indeed, the occurrence of flip and Neimark–Sacker bifurcations implies the possible coexistence of periodic or quasiperiodic attractors even when the interior equilibrium is locally stable. Nevertheless, it remains an interesting question whether global asymptotic stability of the interior fixed point can be established in restricted parameter regimes, for instance under weak eco-evolutionary feedback or small genetic variance. Addressing this problem would require genuinely global techniques, such as the construction of suitable Lyapunov functions or the identification of order-preserving or contractive structures in the dynamics.

I also plan to consider periodically forced environments. Allowing the intrinsic growth rate $b_0$ or other demographic parameters to vary periodically in time would lead to a nonautonomous eco-evolutionary system. Such models are biologically relevant in seasonal environments and are known to generate rich dynamical behavior even in purely ecological settings. Understanding how periodic forcing interacts with evolutionary feedback, and how it modifies the bifurcation structure described here, remains an open and challenging problem.

Another natural extension is to higher-dimensional trait spaces. In many biological applications, adaptation occurs along multiple trait axes, leading to vector-valued evolutionary dynamics. The resulting eco-evolutionary system would involve a higher-dimensional map, where the interaction between population density and multivariate trait evolution could give rise to new instability mechanisms, mode interactions, and complex attractors.

From a mathematical viewpoint, such extensions raise questions about the structure of invariant manifolds, the persistence of invariant tori, and the possible coexistence of multiple attractors. From a biological standpoint, they would allow for a more realistic representation of adaptive processes in structured populations.

I expect that the framework developed in this paper provides a solid basis for addressing these questions and for further exploring the dynamical consequences of eco-evolutionary feedbacks in discrete-time models.