Rate-induced tracking for concave or d-concave transitions in a time-dependent environment with application in ecology

TL;DR

It is shown that finite-time Lyapunov exponents calculated along a locally pullback attractive solution are efficient indicators (early-warning signals) of the presence of a tipping point.

Abstract

This paper investigates biological models that represent the transition equation from a system in the past to a system in the future. It is shown that finite-time Lyapunov exponents calculated along a locally pullback attractive solution are efficient indicators (early-warning signals) of the presence of a tipping point. Precise time-dependent transitions with concave or d-concave variation in the state variable giving rise to scenarios of rate-induced tracking are shown. They are classified depending on the internal dynamics of the set of bounded solutions. Based on this classification, some representative features of these models are investigated by means of a careful numerical analysis.

Figures (7)

• Figure 1: Approximation of finite-time Lyapunov exponent $t\mapsto\lambda_u(c,T,t)$ of the upper local pullback attractor $u_c$ of \ref{['eq:6Bnumericalproblem']}$_c$ for $c=0.999999267212$ and for different values of the length $T$ of the sliding window. In red dashed line, the Lyapunov exponent $L$ of $\tilde{u}_{\gamma_+}$; and in red solid lines, $\kappa\, L$ for $\kappa\in\{0.5,0.8\}$, two possible warning signals of overgrowth of $\lambda_u(c,T,t)$.
• Figure 2: In grey, for different values of $T>0$, the numerical approximation of the region of pairs $(\kappa,c)$ for which there exists $t\in[-400,400]$ such that $\lambda_u(c,T,t)\geq\kappa\, L$, that is, the grey region contains the points for which the EWS calculated with the finite time-Lyapunov exponents of the locally pullback attractive solution $u_c$ of \ref{['eq:6Bnumericalproblem']}$_c$ is issued. In black dashed line, an approximation to the tipping point $c_0$.
• Figure 3: In the first line of pictures, approximation of the locally pullback attractive solutions $l_c<u_c$ (solid red) and the locally pullback repulsive solution $m_c$ (dashed blue) of \ref{['eq:6Bnumericalproblem']}$_c$ for different values of the rate $c>0$. In the second line, approximation of the finite-time Lyapunov exponent $t\mapsto\lambda_u(c,T,t)$ (FTLE) of $u_c$ for a sliding window of $T=50$ units of time and different values of the rate $c$. The red horizontal line represents the warning threshold $\kappa\,L$ for $\kappa=0.6$. The black dashed vertical line represented in both first and second lines of graphs represents the time at which the warning signal is produced, that is, the first $t\in\mathbb{R}$ such that $\lambda_u(c,T,t)\geq\kappa\, L$.
• Figure 4: Illustration of no-return points $u_\Delta(s_3)<m_{c_*}(s_3)$, for any $s_3\geq0$, (left) and safe points $u_\Delta(s_2)>m_{\Delta(s_2)}(s_2)$, for any $s_2>1.15$ (right). In red solid line, the locally pullback attractive solution $u_\Delta$ of \ref{['eq:6Cnonconstant']}, in blue dashed-dotted line, the curve of values of the static \ref{['eq:6constantproblems']}$_{\Delta(t)}$ repellers $t\mapsto m_{\Delta(t)}(t)$ (upper), and, in blue dashed line, the locally pullback repulsive solution of the future $m_{c_*}$ (lower). For both pictures, $f(t,x,\gamma)=r(t)\,x\,(1-x/K(t))-(52-13\gamma)\,x\,/(x+10)$ with $r(t)=2+\sin(t)$ and $K(t)=90+18\sin^2(t\sqrt{5}/2)$, and $\Gamma(t)=-550/(1000+t^2)$. In both cases, $c=19.652326446580$ is the nearest point of CaseA with 12 decimal digits to $c_0$, $t_0=0$ and the warning point $s_1\approx-24.34$ does not appear in the representation. On the left, $\Delta(t)=35-300/(10+t^2)$, on the right $\Delta(t)=30-147/(6+4t^2)$. (Remark 4.7 of Ref. dno3 shows that the strictly increasing character of $\gamma\mapsto f(t,x,\gamma)$ for $(t,x)\in\mathbb{R}\times(0,\infty)$ is enough).
• Figure 5: Rate-induced tracking with time-dependent rate for the concave logistic model \ref{['eq:concave_logistic']} with $\Gamma(\Delta(dt)\,t)=2/\pi \arctan(\Delta(dt)\,t)$ and $I(t)=-\sin(t/2)-\sin(\sqrt5t)+0.895$, and $\Delta(t)=c_-/(1+e^t)+c_+/(1+e^{-t})$, where $c_-=0.25$ and $c_+=0.74$. On the left-hand side there are four panels. The upper two panels show the locally pullback attracting solution $a_\Delta(t)$ in red, and the locally pullback repelling solution $r_\Delta(t)$ in blue in the extended phase space for the values of $d$ equal to $d=1.5$ (left) and $d=3$ (right). The lower panels show the finite-time Lyapunov exponent $t\mapsto \lambda_a(c,T,t)$ (in blue) calculated along the attractor $a_\Delta(t)$ for a sliding window of $T=25$ units of time. In analogy with Figure \ref{['fig:42simulations']} the red horizontal line represents the warning threshold corresponding to $\kappa=0.9$ of the Lyapunov exponent of the attractor in the past-limit problem. In all the first four panels, the black dashed vertical line represents the time at which the warning signal is produced. The central panel shows a numerical approximation of the bifurcation map $\boldsymbol\lambda_*(c,0)$ of \ref{['eq:concave_logistic']} for $c\in[0,1]$; notice that between the chosen $c_-=0.25$ and $c_+=0.74$ there is a whole interval of values for which $\boldsymbol\lambda_*(c)>0$. The right-hand side panel shows the relative position between the attractor $a_{c_-}$ and the repeller $r_{c_+}$ at $t=0$ providing the condition of tracking for sufficiently high rate $d$ thanks to analysis of the switching problem at $t=0$. The change of variables $y=x+\gamma$, for $\gamma>\!\!>0$, transforms the coefficients and hyperbolic solutions into others with biological meaning.

Theorems & Definitions (8)

• Definition 3.1
• Theorem 3.2: Concave saddle-node bifurcation
• Theorem 3.3: D-concave saddle-node bifurcations
• Proposition B.1
• proof
• Proposition B.2
• proof
• Theorem B.3: D-concave bifurcation