Stochastic assessment of temperature–CO<sub>2</sub> causal relationship in climate from the Phanerozoic through modern times

Demetris Koutsoyiannis; Demetris Koutsoyiannis

doi:10.3934/mbe.2024287

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 7: 6560-6602. doi: 10.3934/mbe.2024287

Previous Article Next Article

Research article Special Issues

Stochastic assessment of temperature–CO₂ causal relationship in climate from the Phanerozoic through modern times

Demetris Koutsoyiannis ^,

Department of Water Resources and Environmental Engineering, School of Civil Engineering, National Technical University of Athens, Zographou, Greece

Received: 29 March 2024 Revised: 01 July 2024 Accepted: 03 July 2024 Published: 10 July 2024

As a result of recent research, a new stochastic methodology of assessing causality was developed. Its application to instrumental measurements of temperature (T) and atmospheric carbon dioxide concentration ([CO₂]) over the last seven decades provided evidence for a unidirectional, potentially causal link between T as the cause and [CO₂] as the effect. Here, I refine and extend this methodology and apply it to both paleoclimatic proxy data and instrumental data of T and [CO₂]. Several proxy series, extending over the Phanerozoic or parts of it, gradually improving in accuracy and temporal resolution up to the modern period of accurate records, are compiled, paired, and analyzed. The extensive analyses made converge to the single inference that change in temperature leads, and that in carbon dioxide concentration lags. This conclusion is valid for both proxy and instrumental data in all time scales and time spans. The time scales examined begin from annual and decadal for the modern period (instrumental data) and the last two millennia (proxy data), and reach one million years for the most sparse time series for the Phanerozoic. The type of causality appears to be unidirectional, T→[CO₂], as in earlier studies. The time lags found depend on the time span and time scale and are of the same order of magnitude as the latter. These results contradict the conventional wisdom, according to which the temperature rise is caused by [CO₂] increase.

Keywords:

Citation: Demetris Koutsoyiannis. Stochastic assessment of temperature–CO2 causal relationship in climate from the Phanerozoic through modern times[J]. Mathematical Biosciences and Engineering, 2024, 21(7): 6560-6602. doi: 10.3934/mbe.2024287

Related Papers:

[1]	Qin Zhong, Na Li . Generalized Perron complements in diagonally dominant matrices. AIMS Mathematics, 2024, 9(12): 33879-33890. doi: 10.3934/math.20241616
[2]	Qin Zhong, Ling Li, Gufang Mou . Generalized Perron complements of strictly generalized doubly diagonally dominant matrices. AIMS Mathematics, 2025, 10(6): 13996-14011. doi: 10.3934/math.2025629
[3]	Qin Zhong, Chunyan Zhao . Extended Perron complements of M-matrices. AIMS Mathematics, 2023, 8(11): 26372-26383. doi: 10.3934/math.20231346
[4]	Baifeng Qiu, Zhiping Xiong . The reverse order law for the weighted least square $g$ -inverse of multiple matrix products. AIMS Mathematics, 2023, 8(12): 29171-29181. doi: 10.3934/math.20231494
[5]	Yali Meng . Existence of stable standing waves for the nonlinear Schrödinger equation with attractive inverse-power potentials. AIMS Mathematics, 2022, 7(4): 5957-5970. doi: 10.3934/math.2022332
[6]	Dizhen Ao, Yan Liu, Feng Wang, Lanlan Liu . Schur complement-based infinity norm bounds for the inverse of $S$ -Sparse Ostrowski Brauer matrices. AIMS Mathematics, 2023, 8(11): 25815-25844. doi: 10.3934/math.20231317
[7]	Yuli Zhang, Sizhong Zhou . An $A_{\alpha}$ -spectral radius for the existence of $\{P_3, P_4, P_5\}$ -factors in graphs. AIMS Mathematics, 2025, 10(7): 15497-15511. doi: 10.3934/math.2025695
[8]	Qin Zhong . Some new inequalities for nonnegative matrices involving Schur product. AIMS Mathematics, 2023, 8(12): 29667-29680. doi: 10.3934/math.20231518
[9]	Baijuan Shi . A particular matrix with exponential form, its inversion and some norms. AIMS Mathematics, 2022, 7(5): 8224-8234. doi: 10.3934/math.2022458
[10]	Qi Xiao, Jin Zhong . Characterizations and properties of hyper-dual Moore-Penrose generalized inverse. AIMS Mathematics, 2024, 9(12): 35125-35150. doi: 10.3934/math.20241670

Abstract

1. Introduction

Pest control aims to develop the science, and to design the practice of managing pests, plant diseases and other pest organisms that could have adverse impacts on agricultural production, natural environment, even for our lifestyle ^{[1,2,3,4,5,6]}. In particular, pest outbreaks decrease food production every year, from tomatoes and potatoes to corn. Such as, grape-nematodes can cause immense harvest losses of up to $50\%$ in several crops around the world ^[7], and the caterpillars of spodoptera frugiperda are able to damage more than $180$ species of plants due to they have a very wide host range, and can cause $15$ – $73\%$ reduction in corn production ^[8]. For the above reasons, pest control has been a critical issue to be solved urgently in agriculture and ecology.

Integrated pest management (IPM) ^[9,10,11,12] is an ecosystem-based strategy that focuses on long-term prevention of pests through a comprehensive use of techniques such as biological control ^[13,14,15], habitat manipulation ^[16], modification of cultural practices ^[17,18], chemical control ^[19], and use of resistant varieties ^[20]. This is the most effective and environmentally sensitive approach to pest management which depends on the combination of common-sense practices, at the same time, in the process of pest control, both economical and environmental factors should be considered.

An important factor which influences the IPM strategy is the Threshold Policy Control (TPC) ^[21,22]. This includes two main concepts: The Economic Injury Level (EIL) ^[23] and the Economic Threshold (ET) ^[24,25]. When the pest density level is above EIL, control strategies have to be applied. The ET is the limit pest density that should not be reached. The two concepts have been developed and applied to guide the intervention strategies, to achieve the core goal of IPM, that is, to maintain the pest density below the EIL rather than seeking to eradicate them. Pest management models with threshold control strategies have been widely studied ^{[18,26,27,28,29,30]}, but in most studies, pest population growth is considered to be continuous. For certain species, however, there is no overlap between successive generations, so it is reasonable to consider a discrete model, in that the population has a short life expectancy. In this paper, we formulate a switching host-parasitoid ecosystem with Beverton-Holt growth and threshold control strategies, which consists of pesticide spraying and natural enemy releasing when the pest population reaches the ET. Some theoretical, numerical, biological analyses are given to investigate the effectiveness of the TPC for pest outbreaks.

The organization of the present paper is as follows: in the next section, we propose a switching host-parasitoid ecosystem induced by threshold control strategy. In section 3, by using qualitative analysis techniques related to discrete dynamical systems, we study the threshold conditions which guarantee the existence and stability of equilibria for the two subsystems. In sections 4 and 5, we provide some numerical simulations for the bifurcation and parameter sensitivity analysis respectively. The paper ends with some interesting biological conclusions, which complement the theoretical findings.

2. Switching model formulation

The difference equation

$\begin{equation} \left. \begin{array}{l} H_{t+1} = \frac{aH_t}{b+H_t}, \qquad t\in \mathbb{N} = {0, 1, 2, \cdots}\\ \end{array}\right. \end{equation}$

(2.1)

is the well-known classic Beverton-Holt model ^[31], that describes the population intra-specific competition growth in a single species system. Here $H_t$ stands for the population density of generation $t$ , $a$ is an inherent growth rate parameter (^[32] see for detail) and $b$ is a constant. Model (2.1) is also referred to as the discrete Pielou logistic model ^[33].

In order to describe the inter-specific interactions for two species, a classic discrete-generation host-parasitoid interaction model was developed by Nicholson and Bailey ^[34]. Tang ^[35] extended the Nicholson-Bailey model by including an intergenerational survival rate for the parasitoid

$\begin{equation} \left\{ \begin{array}{l} H_{t+1} = H_t\exp[\alpha-\beta P_t], \\ P_{t+1} = H_t[1-\exp(-\beta P_t)]+\gamma P_t. \end{array}\right. \end{equation}$

(2.2)

In model (2.2), $H_t$ and $P_t$ represent the density of hosts (pests) and parasitoids (natural enemies) at the $t$ -th generation, respectively, $\alpha$ is the intrinsic growth rate for hosts in the absence of parasitoids, $\beta$ denotes the searching efficiency of parasitoids, $\gamma$ is the density-independent survival rate of the parasitoid, and $0 < \gamma < 1$ . The terms $\exp(-\beta P_t)$ and $[1-\exp(-\beta P_t)]$ stand for the probability that a host individual succeeds and fails in escaping from parasitoids, respectively.

In ^[35], the authors investigated the existence and stability of the solutions for Host-parasitoid models with impulsive control at both fixed and unfixed times. Their results indicated that varying dosages and frequencies of insecticide applications, as well as the numbers of parasitoids released, are crucial in pest control. Here, we discuss the host-parasitoid model with Beverton-Holt growth and threshold control strategies.

Based on models (2.1) and (2.2), we establish the following host-parasitoid model with Beverton-Holt growth:

$\begin{equation} \left\{ \begin{array}{l} H_{t+1} = \frac{aH_t}{b+H_t}\exp[\alpha-\beta P_t], \\ P_{t+1} = H_t[1-\exp(-\beta P_t)]+\gamma P_t. \end{array}\right. \end{equation}$

(2.3)

As mentioned in the introduction, the main purpose of IPM is to maintain the pest density below the EIL rather than seeking to eradicate them, by releasing natural enemies or spraying pesticide once the density of pests reaches the ET. This yields the following control model with threshold control strategies

$\begin{equation} \left\{ \begin{array}{l} H_{t+1} = (1-k)\frac{aH_t}{b+H_t}\exp[\alpha-\beta P_t], \\ P_{t+1} = (1+r)\{H_t[1-\exp(-\beta P_t)]+\gamma P_t\}+\tau, \end{array} \right. \end{equation}$

(2.4)

where $k$ stands for the instantaneous killing rate for host population only, $r$ denotes the proportional release rate and $\tau$ is the release rate for parasitoid population, here, $0 < k < 1$ , $0 < r < 1$ and $\tau > 0$ .

As a consequence, combining models (2.3) and (2.4), we propose the following discrete switching host-parasitoid model induced by the threshold control strategies

$\begin{equation} \left\{ \begin{array}{ll} \left. \begin{array}{l} H_{t+1} = \frac{aH_t}{b+H_t}\exp[\alpha-\beta P_t], \\ P_{t+1} = H_t[1-\exp(-\beta P_t)]+\gamma P_t, \end{array} \right\}& H_t \lt ET, \\ \left. \begin{array}{l} H_{t+1} = (1-k)\frac{aH_t}{b+H_t}\exp[\alpha-\beta P_t], \\ P_{t+1} = (1+r)\{H_t[1-\exp(-\beta P_t)]+\gamma P_t\}+\tau, \end{array} \right\}& H_t\geq ET, \\ \end{array}\right. \end{equation}$

(2.5)

where $ET$ is the control threshold, which depends on the crop output value and pest density.

Discrete switching model (2.5) is a dynamical system subject to a threshold policy: IPM control strategies are applied only when $H_t\geq ET$ . Detailed explanation about the threshold policy can be found in ^[21,22].

3. Mathematical analysis for two subsystems

In this section, we consider the dynamical behaviors of the two subsystems of model (2.5).

First, we define two regions as follows:

$\begin{equation*} \begin{array}{c} G_1 = \{(H_t, P_t)| H_t \lt ET, H_t \gt 0, P_t \gt 0, t\in \mathbb{N}\}, \\ G_2 = \{(H_t, P_t)| H_t\geq ET, H_t \gt 0, P_t \gt 0, t\in \mathbb{N}\}, \end{array} \end{equation*}$

and the discrete switching model (2.5) in region $G_1$ (resp. $G_2$ ) as model $F_{G_1}$ (resp. $F_{G_2}$ ).

Furthermore, we investigate the dynamical behaviors of the two subsystems $F_{G_1}$ and $F_{G_2}$ .

Theorem 3.1. If the following condition holds:

$\begin{equation} 1-\gamma \lt \beta [a\exp{\alpha}-b], \end{equation}$

(3.1)

then subsystem $F_{G_1}$ contains a unique internal equilibrium $(H^*_{1}, P^*_{1})$ which is locally asymptotically stable provided that

$\begin{equation} \frac{b+H^*_1\gamma}{b+H^*_1}+\frac{\beta H^*_1(b+2H^*_1)}{a\exp\alpha} \lt 1+\beta H^*_1, \end{equation}$

(3.2)

and

$\begin{equation} \frac{b\gamma}{b+H^*_1}+\beta H^*_1 \lt 1+\frac{\beta (H^*_1)^2}{a\exp\alpha}. \end{equation}$

(3.3)

Proof. We compute the internal equilibrium $(H^*_1, P^*_1)$ of subsystem $F_{G_1}$ by solving the following system of equations:

$\begin{equation} \left\{ \begin{array}{l} H_{1}^* = \frac{aH_1^*}{b+H_1^*}\exp[\alpha-\beta P_1^*], \\ P_{1}^* = H_1^*[1-\exp(-\beta P_1^*)]+\gamma P_1^*. \end{array} \right. \end{equation}$

(3.4)

Through some simple computations, we deduce the following equation:

$\begin{equation} (1-\gamma)P_{1}^* = [a\exp{(\alpha-\beta P_1^*)}-b][1-\exp(-\beta P_1^*)]. \end{equation}$

(3.5)

In order to analyse the above equation, we define two auxiliary functions:

$\left\{ \begin{array}{ll} f_1(x) = (1-\gamma)x, \\ g_1(x) = [a\exp{(\alpha-\beta x)}-b][1-\exp(-\beta x)]. \end{array} \right.$

It is easy to see that $f_1(0) = g_1(0) = 0$ and $\lim\limits_{x\rightarrow +\infty}g_1(x) = -b < 0$ , and $f_1'(0) < g_1'(0)$ from inequality (3.1). These properties assure that there exists an $x^* > 0$ where $f_1$ and $g_1$ intersect. In addition, we find that there is a unique positive solution of $g_1'(x) = 0$ :

$x^* = -\frac{1}{\beta}\ln\frac{a+b\exp(-\alpha)}{2a},$

which implies that Eq (3.5) has a unique positive solution $P_1^*$ . Therefore, we have shown the existence and uniqueness of the internal equilibrium $(H^*_{1}, P^*_{1})$ of subsystem $F_{G_1}$ .

Next, we investigate the local stability of $(H_1^*, P^*_1)$ . We linearize subsystem $F_{G_1}$ around the positive equilibrium $(H_1^*, P^*_1)$ and construct the Jacobian matrix

$\mathcal{J} = \left( \begin{array}{cc} \frac{b}{b+H^*_1} & -\beta H^*_1 \\ 1- \frac{(b+H^*_1)}{a\exp\alpha }& \frac{\beta H^*_1(b+H^*_1)}{a\exp\alpha }+\gamma \\ \end{array} \right).$

The characteristic equation calculated at the internal equilibrium $(H_1^*, P^*_1)$ is given by

$\lambda^2-Trace(\mathcal{J})\lambda +Det(\mathcal{J}) = 0,$

where $Trace(\mathcal{J}) = \frac{b}{b+H^*_1}+\frac{\beta H^*_1(b+H^*_1)}{a\exp\alpha}+\gamma$ , $Det(\mathcal{J}) = \frac{b\gamma}{b+H^*_1}+\beta H^*_1-\frac{\beta {H^*_1}^2}{a\exp\alpha}$ . We see that inequalities (3.2) and (3.3) deduce that $|Trace(\mathcal{J})| < 1+Det(\mathcal{J}) < 2$ , and by Jury criteria ^[36], the local asymptotic stability of the internal equilibirum $(H_1^*, P^*_1)$ is obtained. This completes the proof.

We also obtain the existence and stability of the internal equilibrium of subsystem $F_{G_2}$ .

Theorem 3.2. If $(1+r)\gamma < 1$ , subsystem $F_{G_2}$ contains an internal equilibrium $(H^*_{2}, P^*_{2})$ . It is also locally asymptotically stable if the following conditions are satisfied:

$\frac{b+H^*_2(1+r)\gamma}{b+H^*_2}+\frac{\beta(1+r) H^*_2(b+2H^*_2)}{a(1-k)\exp\alpha } \lt 1+\beta(1+r) H^*_2,$

and

$\frac{b(1+r)\gamma}{b+H^*_2}+\beta(1+r) H^*_2 \lt 1+\frac{\beta(1+r) (H^*_2)^2}{a(1-k)\exp\alpha }.$

The proof of Theorem 3.2 is similar to Theorem 3.1 and we choose to omit it.

The above results suggest that parasitoids and hosts can co-exist under some conditions.

4. Numerical bifurcation analysis

In order to explore the complexity of dynamical behaviors of model (2.5), we perform numerical simulations to show a variety of bifurcation phenomena including one-parameter and multi-parameter bifurcations. The definition and range of parameters are given in Table 1.

Table 1. Parameters in model (2.5).

Parameters	Definitions	Units	Range	References
$\alpha$	Intrinsic growth rate of hosts	–	[1.5, 3.3]	^[37,38]
$\beta$	Searching efficiency of parasitoids	–	[0.01, 3]	^[37,38]
$\gamma$	Density-independent survival rate of parasitoids	Day $^{-1}$	[0.01, 0.5]	^[39]
$a$	Inherent growth rate of hosts	Day $^{-1}$	[0.2, 1.5]	^[40]
$b$	Half-saturation constant	Individual /m $^2$	[0.08, 6]	^[40]
$k$	Instantaneous killing rate of hosts	–	[0.1, 0.9]	Assumed
$r$	Proportional release rate of parasitoids	–	[0.1, 0.7]	Assumed
$\tau$	Release rate for parasitoids	Individual/m $^2$	[0.04, 8]	Assumed
$ET$	Control threshold	Individual/m $^2$	[0.01, 4]	Assumed

| Show Table

DownLoad: CSV

4.1. Equilibrium set for switching system (2.5)

In this subsection, we focus on the set of equilibria for model (2.5) separately in the two different subsystems. First, we refer to the concept of real and virtual equilibria ^[37,41].

Definition 4.1. $E^*(H^*, P^*)$ is a real equilibrium of switching system (2.5) if $E^*$ is an equilibrium of subsystem $F_{G_1}$ (resp. $F_{G_2}$ ) and $H^* < ET$ (resp. $H^*\geq ET$ ). Similarly, $E^*$ is a virtual equilibrium if it is an equilibrium of subsystem $F_{G_1}$ (resp. $F_{G_2}$ ) and $H^*\geq ET$ (resp. $H^* < ET$ ). In the following, we denote by $E^1_{r}$ (resp. $E^2_r$ ) the real equilibrium of subsystem $F_{G_1}$ (resp. $F_{G_2}$ ) and by $E^1_{v}$ (resp. $E^2_v$ ) the virtual equilibrium of subsystem $F_{G_1}$ (resp. $F_{G_2}$ ) when they exist.

In we choose the intrinsic growth rate $\alpha$ of hosts and $ET$ as bifurcation parameters, and we can divide the parameter space into the following six regions according to the number and classification of equilibria:

Figure 1. Bifurcation diagram obtained by varying

$\alpha$ and

$ET$ for the equilibria of model (2.5). The other parameters are

$r = 0.54, a = 0.9, b = 6, \tau = 0.9, \gamma = 0.18, k = 0.3, \beta\in[0, 0.5]$ .

DownLoad: Full-Size Img PowerPoint

Region Ⅰ (blue): No interior equilibria;

Region Ⅱ (green): Only $E_v^1$ exists;

Region Ⅲ (yellow): Only $E_v^2$ exists;

Region Ⅳ (cyan): Only $E_r^2$ exists;

Region Ⅴ (magenta): $E_v^1$ and $E_v^2$ coexist;

Region Ⅵ (red): $E_v^1$ and $E_r^2$ coexist.

For an optimal pest control, it is necessary to design an effective control to keep the density of pest population below the ET. This requires us to choose the appropriate parameters $\alpha$ , $\beta$ and $ET$ such that the internal equilibria of both subsystems $F_{G_1}$ and $F_{G_2}$ become virtual. So the optimal control parameter regions are Ⅱ and Ⅴ in this case.

4.2. Bifurcation analyses and chaos

In this subsection, we study how relevant control parameters, such as the killing rate $k$ of hosts and the constant releasing rate $\tau$ of parasitoids, affect the dynamics of model (2.5) using numerical simulations.

We first choose the killing rate $k$ of the host population as the bifurcation parameter and fix the other parameters as in . It turns out that the choice of $k$ is critical to study some more complicated dynamics of system (2.5), especially for $k\in[0.6, 0.9]$ . In particular, chaos appears as $k\in [0.715, 0.762], [0.808, 0.814], [0.825, 0.875]$ and $[0.895, 0.9]$ . Furthermore, we can observe period-doubling and periodic-halving bifurcations as $k$ increases from 0.6 to 0.65.

Figure 2. Bifurcation analyses with

$k$ as bifurcation parameter. The other parameters are

$\alpha = 3.3, \tau = 0.04, \beta = 1, \gamma = 0.08, a = 0.6, b = 0.08, r = 0.7, ET = 2.5$ and

$(H_0, P_0) = (3, 2)$ .

DownLoad: Full-Size Img PowerPoint

We then study the bifurcation diagram of model (2.5) with respect to the releasing rate $\tau$ of the parasitoid population. As $\tau$ increases from $0.2$ to $1.8$ in , the switching host-parasitoid system experiences some complex and interesting phenomena such as periodic-doubling, periodic-adding, periodic-halving and periodic window bifurcations, chaos and so on. The dynamical behavior of model (2.5) is particularly sensitive to $\tau$ , which makes it be a relevant pest control parameter.

Figure 3. Bifurcation analyses with

$\tau$ as bifurcation parameter. The other parameters are

$\alpha = 3, \beta = 1.45, \gamma = 0.02, a = 1, b = 1.2, r = 0.48, ET = 2, k = 0.9$ and

$(H_0, P_0) = (2, 3)$ .

DownLoad: Full-Size Img PowerPoint

At last, we study the effect of the searching efficiency $\beta$ of the parasitoid population on the dynamics of switching model (2.5). It shows that several complex dynamical behaviours arise also in this case (Figure 4). Addressing these complexities is one of the main challenges for pest control, which requires us to focus on the interaction between host and parasitoid populations, and to design effective control strategies in accordance with IPM's goal.

Figure 4. Bifurcation analyses with

$\beta$ as bifurcation parameter. The other parameters are

$\alpha = 1.5, \tau = 1.2, \gamma = 0.01, a = 1.5, b = 1.2, r = 0.1, ET = 4, k = 0.6$ and

$(H_0, P_0) = (3, 5)$ .

DownLoad: Full-Size Img PowerPoint

5. Sensitivity analysis of the initial values

Interaction between hosts and parasitoids is a key to limit pest spread and to control the dynamics of both populations. Especially, the final state of model (2.5) is extremely sensitive to initial densities, hence this section we investigate how the initial values affect the switching frequencies, and analyse the cases of coexistence of multiple attractors in the proposed switching system.

5.1. Switching frequency and control strategy

It is relevant to introduce the following definition of switching frequency ^[37,41].

Definition 5.1. If $(H_t-ET)(H_{t+1}-ET)\leq 0$ and $H_{t+1}\neq ET$ hold, then we say that switching system (2.5) experiences one time switch, where $t$ is a switch-point. An interval between two continuous switch-points is known as the switching frequency.

As shown in Figure 5, different initial values can lead to different stable or unstable states for the switching frequencies. That is, Figure 5A, B show unstable switching frequencies, while Figure 5C shows stable ones, and the frequency shown in Figure 5C is higher than the other two.

Figure 5. Switching frequency of system (2.5) with

$(H_0, P_0) = (3.0, 2.5)$ ,

$(3.5, 2.2)$ ,

$(4.0, 3.5)$ from [A] to [C]. Parameters are

$a = 1.5, b = 0.2, \beta = 0.16, \tau = 0.9, \gamma = 0.6, k = 0.2, r = 0.2, \alpha = 1.7, ET = 0.8$ .

DownLoad: Full-Size Img PowerPoint

We know that switching frequency plays an important role in pest control, since a high switching frequency requires strong control measures which include the use of resources, such as pesticides, labor force, equipments, and so on. However, in the real world, it could be difficult to have all available resources for economical or environmental reasons. Therefore it is necessary to keep everything into account in designing an optimal initial value for pest control.

Next, we analyse the phase diagram of host-parasitoid densities for different initial values. For a given threshold $ET$ , Figure 6 describes how the dynamics of model (2.5) changes as the initial values do with four different cases. In particular, Figure 6A shows a situation that does not require control strategies, while – require respectively $1, 2$ and several applications of IPM strategies.

Figure 6. The relationships between switching frequency and initial value. Parameters are

$a = 0.2, b = 0.3, \beta = 3, \tau = 0.2, \gamma = 0.3, k = 0.1, r = 0.1, \alpha = 2$ .

DownLoad: Full-Size Img PowerPoint

In addition, we discuss how the initial densities affect the pest outbreak frequency. In , the initial values of host and parasitoid populations are divided into five regions denoted by Ⅰ (green), Ⅱ (yellow), Ⅲ (magenta), Ⅳ (red) and Ⅴ (blue), which depend on the number of outbreaks that model (2.5) has to face (respectively $0, 1, 2, 3$ and $4$ or more outbreaks). As expected, the choice of initial values in Region Ⅰ is the most favorable for pest control since it does not require any strategy, while initial values in Region Ⅴ can make pest control problematic, even can have a negative economical and environmental impact.

Figure 7. Basin region for the initial densities of host and parasitoid populations, parameters are identical with those in Figure 6.

DownLoad: Full-Size Img PowerPoint

5.2. Multiple attractors and their coexistence

We have already discussed multiple attractors and their coexistence in the previous section, so here we focus on how initial densities affect these.

Figure 8 describes three host-outbreak periodic attractors with different amplitudes and periods for three choices of initial values. In particular, the period and amplitude of the host population of the third attractor is smaller, while the amplitude of the parasitoid population is similar in all attractors, but is slightly larger in the second case. In addition, the period is the largest in the first case which is seven generations.

Figure 8. Three coexisting attractors of system (2.5) with

$(H_0, P_0) = (1.5, 3.0)$ ,

$(1.5, 3.5)$ ,

$(1.5, 4.0)$ from top to bottom. Other parameters are

$a = 1.5, b = 0.2, \beta = 0.16, \tau = 8, \gamma = 0.5, k = 0.2, r = 0.2, \alpha = 1.6, ET = 0.6$ .

DownLoad: Full-Size Img PowerPoint

To further investigate the role of initial values in the dynamics of model (2.5), we study the basin of attraction of these three host-outbreak solutions in Figure 8. The parameter space of initial densities is divided into three regions (blue, green and red) shown in Figure 9, which correspond to the attractors from top to bottom of Figure 8. We see that choices of initial values in the red area may be ideal for pest control since this attractor has a smaller host amplitude than the other two. This study shows that it is necessary to understand the initial values of both hosts and parasitoids for a successful pest control.

Figure 9. [A] Basin of attraction of the attractors shown in Figure 8; [B] The magnification of [A] around small host-parasitoid initial densities.

DownLoad: Full-Size Img PowerPoint

6. Discussion

Pest control is an essential task of the agricultural and biological managements, which mainly involves chemical control through pesticide spraying and biological control through natural enemy releasing. In order to find the optimal time and dosage of these control methods, the threshold policy (TP) is introduced in the IPM strategies. Based on this, we propose a novel discrete switching ecosystem with threshold strategy, and do some mathematical, numerical and biological analyses to verify the effectiveness of the model in pest control.

The proposed switching model is highly sensitive to the choice of the threshold value $ET$ , since it is the minimum pest density above which the chemical and biological control measures should be applied. The introduction of model (2.5) is consistent with the main purpose of IPM which is to maintain the pest density below the EIL instead of eradicating them completely.

We apply qualitative analysis techniques related to difference equations to study the existence and stability of equilibria for subsystems $F_{G_1}$ and $F_{G_2}$ . The existence of a large number of possible equilibria for model (2.5) leads to the possibility of single and multiple-parameter bifurcations and chaos, then we show this complexity of dynamical behaviors of the switching ecosystem (2.5) through some numerical simulations.

Specially, the multi-parameter bifurcation diagram divides the space into regions by number of possible equilibria and their classification, while the single parameter bifurcation diagrams reveal the existence of periodic and chaotic solutions, period-doubling, periodic-halving, periodic window bifurcations, and so on. Both diagrams show how some key control parameters, such as the killing rate $k$ of $H_t$ and the constant releasing rate $\tau$ of $P_t$ , affect the dynamics of model (2.5), therefore it is crucial to choose suitable control parameters for pest control.

In general, sensitivity analysis of initial value is a relevant tool to analyse the model numerically. In this paper, we focus on the initial state of host and parasitoid populations, and investigate how initial values affect the switching frequencies and the coexistence of multiple attractors of model (2.5) in Figures 5–9. Therefore, initial densities of host and parasitoid populations are essential for pest control, thus a complete understanding for the initial interaction of both populations would be a key for pest control.

In this paper, we only focus on the importance of threshold and IPM strategies in pest control. however there are several other factors including stochasticity in the environment, residual effects on hosts of pesticides and limited resources that can affect pest control in real life. Therefore it is of great theoretical and practical significance to introduce these factors into pest control model in future studies.

Acknowledgments

The authors would like to express their gratitude to Professor Jianhong Wu for his kind suggestions, and also to the referees for their helpful comments. This work was supported by the National Natural Science Foundation of China (Grant No. 11601268) and the Scientific Research Program Funded by Shaanxi Provincial Education Department(Grant No. 18JK0336).

Conflict of interest

All authors declare no conflicts of interest in this paper.

References

[1]	D. Koutsoyiannis, C. Onof, A. Christofides, Z. W. Kundzewicz, Revisiting causality using stochastics: 1. Theory, Proc. R. Soc. A, 478 (2022), 20210836. https://doi.org/10.1098/rspa.2021.0835 doi: 10.1098/rspa.2021.0835
[2]	J. Veizer, Celestial climate driver: A perspective from four billion years of the carbon cycle, Geosci. Canada, 32 (2005), 13–28. https://doi.org/10.1142/9789812773890_0010 doi: 10.1142/9789812773890_0010
[3]	J. Veizer, The role of water in the fate of carbon dioxide: implications for the climate system, in 43rd Int. Seminar on Nuclear War and Planetary Emergencies (2011), R. Ragaini (Ed.). World Scientific, 313–327. https://doi.org/10.1142/8232
[4]	J. Veizer, Planetary temperatures/climate across geological time scales, in International Seminar on Nuclear War and Planetary Emergencies—44th Session: The Role of Science in the Third Millennium (2012), 287–288. https://doi.org/10.1142/9789814415019_0023
[5]	J. Laskar, A numerical experiment on the chaotic behaviour of the Solar System, Nature, 338 (1989), 237–238. https://doi.org/10.1038/338237a0 doi: 10.1038/338237a0
[6]	J. Laskar, The limits of Earth orbital calculations for geological time-scale use, Philos. Tr. R. Soc. Lond. A, 357 (1999), 1735–1759. https://doi.org/10.1098/rsta.1999.0399 doi: 10.1098/rsta.1999.0399
[7]	M. J. Duncan, Orbital stability and the structure of the solar system, in 1994, in: Circumstellar Dust Disks and Planet Formation, Proceedings of the 10th IAP Astrophysics Meeting, Institut d'Astrophysique de Paris (1994), edited by: R. Ferlet, and A. Vidal-Madjar, Editions Frontiers, Gif sur Yvette, France, 245–256.
[8]	J. L. Lissauer, Chaotic motion in the Solar System, Rev. Mod. Phys., 71 (1999), 835–845. https://doi.org/10.1103/RevModPhys.71.835 doi: 10.1103/RevModPhys.71.835
[9]	D. Koutsoyiannis, A random walk on water, Hydrol. Earth Syst. Sci., 14 (2010), 585–601. https://doi.org/10.5194/hess-14-585-2010 doi: 10.5194/hess-14-585-2010
[10]	M. Milanković, Nebeska Mehanika, Beograd, 1935.
[11]	M.Milanković, Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeitenproblem, Ko niglich Serbische Akademice, Beograd, 1941.
[12]	M. Milanković, Canon of Insolation and the Ice-Age Problem, Agency for Textbooks, Belgrade, 1998.
[13]	J. D. Hays, J. Imbrie, J., N. J. Shackleton, Variations in the Earth's Orbit: Pacemaker of the Ice Ages, Science, 194 (1976), 1121–1132. https://doi.org/10.1126/science.194.4270.1121 doi: 10.1126/science.194.4270.1121
[14]	G. Roe, In defense of Milankovitch. Geophys. Res. Lett. 33 (2006). https://doi.org/10.1029/2006GL027817 doi: 10.1029/2006GL027817
[15]	N. J. Shaviv, H. Svensmark, J. Veizer, The phanerozoic climate. Ann. N. Y. Acad. Sci., 1519 (2023), 7–19. https://doi.org/10.1111/nyas.14920 doi: 10.1111/nyas.14920
[16]	D. Koutsoyiannis, Relative importance of carbon dioxide and water in the greenhouse effect: Does the tail wag the dog? Preprints, 2024040309 (2024). https://doi.org/10.20944/preprints202404.0309.v1 doi: 10.20944/preprints202404.0309.v1
[17]	D. Koutsoyiannis, Time's arrow in stochastic characterization and simulation of atmospheric and hydrological processes, Hydrol. Sci. J., 64 (2019), 1013–1037. https://doi.org/10.1080/02626667.2019.1600700 doi: 10.1080/02626667.2019.1600700
[18]	D. Koutsoyiannis, Revisiting the global hydrological cycle: is it intensifying?, Hydrol. Earth Syst. Sci., 24 (2020), 3899–3932. https://doi.org/10.5194/hess-24-3899-2020 doi: 10.5194/hess-24-3899-2020
[19]	D. Koutsoyiannis, Z. W. Kundzewicz, Atmospheric temperature and CO₂: Hen-or-egg causality?, Sci, 2 (2020), 83. https://doi.org/10.3390/sci2040083 doi: 10.3390/sci2040083
[20]	D. Koutsoyiannis, Rethinking climate, climate change, and their relationship with water, Water, 13 (2021), 849. https://doi.org/10.3390/w13060849 doi: 10.3390/w13060849
[21]	D. Koutsoyiannis, C. Onof, A. Christofides, Z. W. Kundzewicz, Revisiting causality using stochastics: 2. Applications, Proc. R. Soc. A, 478 (2022), 20210835. https://doi.org/10.1098/rspa.2021.0835 doi: 10.1098/rspa.2021.0835
[22]	D. Koutsoyiannis, C. Onof, Z. W. Kundzewicz, A. Christofides, On hens, eggs, temperatures and CO₂: Causal links in Earth's atmosphere, Sci, 5 (2023), 35. https://doi.org/10.3390/sci5030035 doi: 10.3390/sci5030035
[23]	D. Koutsoyiannis, C. Vournas, Revisiting the greenhouse effect—a hydrological perspective, Hydrol. Sci. J., 69 (2024), 151–164. https://doi.org/10.1080/02626667.2023.2287047 doi: 10.1080/02626667.2023.2287047
[24]	D. Koutsoyiannis, Net isotopic signature of atmospheric CO₂ sources and sinks: No change since the Little Ice Age, Sci, 6 (2024), 17. https://doi.org/10.3390/sci6010017 doi: 10.3390/sci6010017
[25]	J. D. Shakun, P. U. Clark, F. He, S. A. Marcott, A. C. Mix, Z. Liu, et al., Global warming preceded by increasing carbon dioxide concentrations during the last deglaciation, Nature, 484 (2012), 49–54. https://doi.org/10.1038/nature10915 doi: 10.1038/nature10915
[26]	J. C. Beeman, L. Gest, F. Parrenin, D. Raynaud, T. J. Fudge, C. Buizert, et al., Antarctic temperature and CO₂: near-synchrony yet variable phasing during the last deglaciation, Clim. Past, 15 (2019), 913–926. https://doi.org/10.5194/cp-15-913-2019 doi: 10.5194/cp-15-913-2019
[27]	F. Parrenin, V. Masson-Delmotte, P. Köhler, D. Raynaud, D. Paillard, J. Schwander, et al., Synchronous change of atmospheric CO₂ and Antarctic temperature during the last deglacial warming, Science 339, (2013), 1060–1063. https://doi.org/10.1126/science.1226368 doi: 10.1126/science.1226368
[28]	W. Soon, Implications of the secondary role of carbon dioxide and methane forcing in climate change: past, present, and future. Phys. Geogr. 28 (2007), 97–125. https://doi.org/10.2747/0272-3646.28.2.97 doi: 10.2747/0272-3646.28.2.97
[29]	R. A. Berner, Z. Kothavala, GEOCARB Ⅲ: A revised model of atmospheric CO₂ over Phanerozoic time, Am. J. Sci., 301 (2001), 182–204. https://doi.org/10.2475/ajs.301.2.182 doi: 10.2475/ajs.301.2.182
[30]	J. Veizer, Y. Godderis, L. M. François, Evidence for decoupling of atmospheric CO₂ and global climate during the Phanerozoic eon, Nature, 408 (2000), 698–701. https://doi.org/10.1038/35047044 doi: 10.1038/35047044
[31]	N. Caillon, J. P. Severinghaus, J. Jouzel, J. M. Barnola, J. Kang, V. Y. Lipenkov, Timing of atmospheric CO₂ and Antarctic temperature changes across Termination Ⅲ, Science 299 (2003), 1728–1731. https://doi.org/10.1126/science.1078758 doi: 10.1126/science.1078758
[32]	J. B. Pedro, S. O. Rasmussen, T. D. van Ommen, Tightened constraints on the time-lag between Antarctic temperature and CO₂ during the last deglaciation, Clim. Past, 8 (2012), 1213–1221. https://doi.org/10.5194/cp-8-1213-2012 doi: 10.5194/cp-8-1213-2012
[33]	O. Humlum, K. Stordahl, J. E. Solheim, The phase relation between atmospheric carbon dioxide and global temperature, Glob. Planet. Change, 100 (2013), 51–69. https://doi.org/10.1016/j.gloplacha.2012.08.008 doi: 10.1016/j.gloplacha.2012.08.008
[34]	C. R. Scotese, H. Song, B. J. Milels, D. G. van der Meer, Phanerozoic paleotemperatures: The earth's changing climate during the last 540 million years, Earth Sci. Rev., 215 (2021), 103503. https://doi.org/10.1016/j.earscirev.2021.103503 doi: 10.1016/j.earscirev.2021.103503
[35]	E. L. Grossman, M. M. Joachimski, Ocean temperatures through the Phanerozoic reassessed, Sci. Rep., 12 (2022), 8938. https://doi.org/10.1038/s41598-022-11493-1 doi: 10.1038/s41598-022-11493-1
[36]	H. Song, P. B. Wignall, H. Song, X. Dai, D. Chu, Seawater temperature and dissolved oxygen over the past 500 million years, J. Earth Sci., 30 (2019), 236–243. https://doi.org/10.1007/s12583-018-1002-2 doi: 10.1007/s12583-018-1002-2
[37]	J. P. Klages, U. Salzmann, T. Bickert, C.-D. Hillenbrand, K. Gohl, G. Kuhn, et al., Temperate rainforests near the South Pole during peak Cretaceous warmth, Nature 580 (2020), 81–86. https://doi.org/10.1038/s41586-020-2148-5 doi: 10.1038/s41586-020-2148-5
[38]	J. M. Schaefer, R. C. Finkel, G. Balco, R. B. Alley, M. W. Caffee, J. P. Briner, et al., Greenland was nearly ice-free for extended periods during the Pleistocene, Nature, 540 (2016), 252–255. https://doi.org/10.1038/nature20146 doi: 10.1038/nature20146
[39]	K. H. Kjær, M. W. Pedersen, B. De Sanctis, B. De Cahsan, T. S. Korneliussen, C. S. Michelsen, et al., A 2-million-year-old ecosystem in Greenland uncovered by environmental DNA, Nature, 612 (2022), 283–291. https://doi.org/10.1038/s41586-022-05453-y doi: 10.1038/s41586-022-05453-y
[40]	R. A. Berner, GEOCARBSULF: A combined model for Phanerozoic atmospheric O₂ and CO₂, Geochim. Cosmochim. Acta, 70 (2006), 5653–5664. https://doi.org/10.1016/j.gca.2005.11.032 doi: 10.1016/j.gca.2005.11.032
[41]	D. L. Royer, Y. Donnadieu, J. Park, J. Kowalczyk, Y. Godderis, Error analysis of CO₂ and O₂ estimates from the long-term geochemical model GEOCARBSULF, Am. J. Sci., 314 (2014), 1259–1283. https://doi.org/10.2475/09.2014.01 doi: 10.2475/09.2014.01
[42]	G. L. Foster, D. L. Royer, D. J. Lunt, Future climate forcing potentially without precedent in the last 420 million years, Nat. Commun., 8 (2017), 14845. https://doi.org/10.1038/ncomms14845 doi: 10.1038/ncomms14845
[43]	D. L. Royer, Atmospheric CO₂ and O₂ during the Phanerozoic: Tools, patterns and impacts. In Treatise on Geochemistry, 2nd ed.; H. Holland, K. Turekian, Eds., Elselvier, Amsterdam, The Netherlands, (2014), 251–267. https://doi.org/10.1016/B978-0-08-095975-7.01311-5
[44]	W. J. Davis, The relationship between atmospheric carbon dioxide concentration and global temperature for the last 425 million years, Climate, 5 (2017), 76. https://doi.org/10.3390/cli5040076 doi: 10.3390/cli5040076
[45]	R. A. Berner, Addendum to "Inclusion of the Weathering of Volcanic Rocks in the GEOCARBSULF Model" (R. A, Berner, 2006, V. 306, p. 295–302), Am. J. Sci., 308 (2008), 100–103. https://doi.org/10.2475/01.2008.04 doi: 10.2475/01.2008.04
[46]	J. J. Sepkoski, A factor analytic description of the Phanerozoic marine fossil record, Paleobiology, 7 (1981), 36–53. https://doi.org/10.1017/S0094837300003778 doi: 10.1017/S0094837300003778
[47]	T. Westerhold, N. Marwan, A. J. Drury, D. Liebrand, C. Agnini, E. Anagnostou, et al., An astronomically dated record of Earth's climate and its predictability over the last 66 million years, Science, 369 (2020), 1383–1387. https://doi.org/10.1126/science.aba6853 doi: 10.1126/science.aba6853
[48]	A. L. Melott, R. K. Bambach, Analysis of periodicity of extinction using the 2012 geological timescale, Paleobiology, 40 (2014), 177–196. https://doi.org/10.1666/13047 doi: 10.1666/13047
[49]	W. J. Davis, Mass extinctions and their relationship with atmospheric carbon dioxide concentration: Implications for Earth's future, Earth's Future, 11 (2023), e2022EF003336. https://doi.org/10.1029/2022EF003336 doi: 10.1029/2022EF003336
[50]	J. W. Rae, Y. G. Zhang, X. Liu, G. L. Foster, H. M. Stoll, R. D. Whiteford, Atmospheric CO₂ over the past 66 million years from marine archives, Ann. Rev. Earth Planet. Sci., 49 (2021), 609–641. https://doi.org/10.1146/annurev-earth-082420-063026 doi: 10.1146/annurev-earth-082420-063026
[51]	S. Epstein, R. Buchsbaum, H. A. Lowenstam, H. C. Urey, Revised carbonate-water isotopic temperature scale, Geolog. Soc. Am. Bullet., 64 (1953), 1315–1326. https://doi.org/10.1130/0016-7606(1953)64[1315:RCITS]2.0.CO; 2 doi: 10.1130/0016-7606(1953)64[1315:RCITS]2.0.CO; 2
[52]	J. Jouzel, C. Lorius, J. R. Petit, C. Genthon, N. I. Barkov, V. M. Kotlyakov, et al., Vostok ice core: A continuous isotope temperature record over the last climatic cycle (160 000 years), Nature, 329 (1987), 403–408. https://doi.org/10.1038/329403a0 doi: 10.1038/329403a0
[53]	J. R. Petit, J. Jouzel, D. Raynaud, N. I. Barkov, J.-M. Barnola, I. Basile, et al., Climate and atmospheric history of the past 420,000 years from the Vostok ice core, Antarctica, Nature 399 (1999), 429–436. https://doi.org/10.1038/20859 doi: 10.1038/20859
[54]	J.-M. Barnola, P. Pimienta, D. Raynaud, Y. S. Korotkevich, CO₂-climate relationship as deduced from the Vostok ice core: A re-examination based on new measurements and on a re-evaluation of the air dating, Tellus B Chem. Phys, Meteorol., 43 (1991), 83–90. https://doi.org/10.3402/tellusb.v43i2.15249 doi: 10.3402/tellusb.v43i2.15249
[55]	B. Christiansen, F. C. Ljungqvist, Challenges and perspectives for large-scale temperature reconstructions of the past two millennia, Rev. Geoph., 55 (2017), 40–96. https://doi.org/10.1002/2016RG000521 doi: 10.1002/2016RG000521
[56]	A. Moberg, D. M. Sonechkin, K. Holmgren, N. M. Datsenko, W. Karlén, Highly variable Northern Hemisphere temperatures reconstructed from low-and high-resolution proxy data, Nature, 433 (2005), 613–617. https://doi.org/10.1038/nature03265 doi: 10.1038/nature03265
[57]	C. Loehle, J. H. McCulloch, Correction to: A 2000-year global temperature reconstruction based on non-tree ring proxies, Energy Environ., 19 (2008), 93–100. https://doi.org/10.1260/095830508783563109 doi: 10.1260/095830508783563109
[58]	B. Christiansen, F. C. Ljungqvist, The extra-tropical Northern Hemisphere temperature in the last two millennia: reconstructions of low-frequency variability, Clim. Past, 8 (2012), 765–786. https://doi.org/10.5194/cp-8-765-2012 doi: 10.5194/cp-8-765-2012
[59]	A. Indermühle, T. F. Stocker, F. Joos, H. Fischer, H. J. Smith, M. Wahlen, et al., Holocene carbon-cycle dynamics based on CO₂ trapped in ice at Taylor Dome, Antarctica, Nature, 398 (1999), 121–126. https://doi.org/10.1038/18158 doi: 10.1038/18158
[60]	D. M. Etheridge, L. P. Steele, R. L. Langenfelds, R. J. Francey, J.-M. Barnola, V. I. Morgan, Natural and anthropogenic changes in atmospheric CO₂ over the last 1000 years from air in Antarctic ice and firn, J. Geophys. Res. Atmosph. 101, (1996), 4115–4128. https://doi.org/10.1029/95JD03410 doi: 10.1029/95JD03410
[61]	R. J. Francey, C. E. Allison, D. M. Etheridge, C. M. Trudinger, I. G. Enting, M. Leuenberger, et al., A 1000-year high precision record of δ¹³C in atmospheric CO₂, Tellus B, 51 (1999), 170–193. https://doi.org/10.3402/tellusb.v51i2.16269 doi: 10.3402/tellusb.v51i2.16269
[62]	F. Böhm, A. Haase-Schramm, A. Eisenhauer, W. C. Dullo, M. M. Joachimski, H. Lehnert, et al., Evidence for preindustrial variations in the marine surface water carbonate system from coralline sponges, Geochem, Geophys. Geosyst., 3 (2002), 1–13. https://doi.org/10.1029/2001GC000264 doi: 10.1029/2001GC000264
[63]	L. L. R. Kouwenberg, Application of conifer needles in the reconstruction of Holocene CO₂ levels, Ph.D. thesis, Utrecht, Laboratory of Paleobotany and Palynology (LPP) Foundation, LPP Contributions series 16,130 p., 2004. https://dspace.library.uu.nl/bitstream/handle/1874/243/full.pdf (accessed 2024-03-10).
[64]	L. Kouwenberg, R. Wagner, W. Kürschner, H. Visscher, Atmospheric CO₂ fluctuations during the last millennium reconstructed by stomatal frequency analysis of Tsuga heterophylla needles, Geology, 33 (2005), 33–36. https://doi.org/10.1130/G20941.1 doi: 10.1130/G20941.1
[65]	T. B. van Hoof, F. Wagner-Cremer, W. M. Kürschner, H. Visscher, A role for atmospheric CO₂ in preindustrial climate forcing, Proc. Nat. Acad. Sci., 105 (2008), 15815–15818. https://doi.org/10.1073/pnas.0807624105 doi: 10.1073/pnas.0807624105
[66]	J. M. Barnola, M. Anklin, J. Porcheron, D. Raynaud, J. Schwander, B. Stauffer, CO₂ evolution during the last millennium as recorded by Antarctic and Greenland ice, Tellus B Chem. Phys. Meteorol., 47 (1995), 264–272. https://doi.org/10.3402/tellusb.v47i1-2.16046 doi: 10.3402/tellusb.v47i1-2.16046
[67]	M. Anklin, J. Schwander, B. Stauffer, J. Tschumi, A. Fuchs, J. M. Barnola, et al., CO₂ record between 40 and 8 kyr BP from the Greenland Ice Core Project ice core, J. Geophys. Res. Oceans, 102 (1997), 26539–26545. https://doi.org/10.1029/97JC00182 doi: 10.1029/97JC00182
[68]	J. C. McElwain, F. E. Mayle, D. J. Beerling, Stomatal evidence for a decline in atmospheric CO₂ concentration during the Younger Dryas stadial: A comparison with Antarctic ice core records, J. Quaternary Sci., 17 (2002), 21–29. https://doi.org/10.1002/jqs.664 doi: 10.1002/jqs.664
[69]	F. Wagner, L. L. Kouwenberg, T. B. van Hoof, H. Visscher, Reproducibility of Holocene atmospheric CO₂ records based on stomatal frequency, Quaternary Sci. Rev., 23 (2004), 1947–1954. https://doi.org/10.1016/j.quascirev.2004.04.003 doi: 10.1016/j.quascirev.2004.04.003
[70]	C. A. Jessen, M. Rundgren, S. Björck, D. Hammarlund, Abrupt climatic changes and an unstable transition into a late Holocene Thermal Decline: A multiproxy lacustrine record from southern Sweden, J. Quaternary Sci., 20 (2005), 349–362. https://doi.org/10.1002/jqs.921 doi: 10.1002/jqs.921
[71]	M. Steinthorsdottir, B. Wohlfarth, M. E. Kylander, M. Blaauw, P. J. Reimer, Stomatal proxy record of CO₂ concentrations from the last termination suggests an important role for CO₂ at climate change transitions, Quaternary Sci. Rev., 68 (2013), 43–58. https://doi.org/10.1016/j.quascirev.2013.02.003 doi: 10.1016/j.quascirev.2013.02.003
[72]	Y. Wang, A. Momohara, N. Wakamatsu, T. Omori, M. Yoneda, M. Yang, Middle and Late Holocene altitudinal distribution limit changes of Fagus crenata forest, Mt. Kurikoma, Japan indicated by stomatal evidence, Boreas, 49 (2020), 718–729. https://doi.org/10.1111/bor.12463 doi: 10.1111/bor.12463
[73]	S. Azharuddin, P. Govil, T. B. Chalk, M. Shekhar, G. L. Foster, R. Mishra, Abrupt upwelling and CO₂ outgassing episodes in the north-eastern Arabian Sea since mid-Holocene, Sci. Rep., 12 (2022), 3830. https://doi.org/10.1038/s41598-022-07774-4 doi: 10.1038/s41598-022-07774-4
[74]	B. Christiansen, F. C. Ljungqvist, Northern Hemisphere extratropical 2000 year temperature reconstruction, IGBP PAGES/World Data Center for Paleoclimatology Data Contribution Series #2012-049, NOAA/NCDC Paleoclimatology Program, Boulder CO, USA, 2012. https://www.ncdc.noaa.gov/paleo/study/12902 (accessed on 16 March 2024)
[75]	ERA5: data documentation - Copernicus Knowledge Base - ECMWF Confluence Wiki. Available online: https://confluence.ecmwf.int/display/CKB/ERA5%3A+data+documentation #heading-Relatedarticles (accessed on 25 March 2023).
[76]	C. D. Keeling, S. C. Piper, T. P. Whorf, R. F. Keeling, Evolution of natural and anthropogenic fluxes of atmospheric CO₂ from 1957 to 2003, Tellus B: Chem. Phys. Meteorol., 63 (2011), 1–22. https://doi.org/10.1111/j.1600-0889.2010.00507.x doi: 10.1111/j.1600-0889.2010.00507.x
[77]	World Economic Forum, CO₂ levels in atmosphere are at their highest in 800,000 years https://www.weforum.org/agenda/2018/05/earth-just-hit-a-terrifying-milestone-for-the-first-time-in-more-than-800-000-years/(accessed on 16 March 2024
[78]	The Conversation, The three-minute story of 800,000 years of climate change with a sting in the tail, https://theconversation.com/the-three-minute-story-of-800-000-years-of-climate-change-with-a-sting-in-the-tail-73368 (accessed on 16 March 2024)
[79]	NASA, Graphic: The relentless rise of carbon dioxide, https://science.nasa.gov/resource/graphic-the-relentless-rise-of-carbon-dioxide (accessed on 16 March 2024).
[80]	M. Shan, Analysis on the influence of doubled carbon dioxide on the extreme weather, Proceedings of the 3rd International Conference on Materials Chemistry and Environmental Engineering, Applied and Computational Engineering, 3 (2023), 368–373. https://doi.org/10.54254/2755-2721/3/20230554 doi: 10.54254/2755-2721/3/20230554
[81]	H. S. Kang, S. Chester, C. Meneveau, Decaying turbulence in an active-grid-generated flow and comparisons with large-eddy simulation, J. Fluid Mech., 480 (2003), 129–160. https://doi.org/10.1017/S0022112002003579 doi: 10.1017/S0022112002003579
[82]	Center for Environmental and Applied Fluid Mechanics, 30 data files at x/M = 20 for probe separations corresponding to filter scale 5 mm. http://pages.jh.edu/~cmeneve1/datasets/Activegrid/M20/H1 (accessed on 16 March 2024)
[83]	D. Koutsoyiannis, Hydrology and Change, Hydrol. Sci. J., 58 (2013), 1177–1197. https://doi.org/10.1080/02626667.2013.804626 doi: 10.1080/02626667.2013.804626
[84]	D. Koutsoyiannis, Entropy production in stochastics, Entropy, 19 (2017), 581. https://doi.org/10.3390/e19110581 doi: 10.3390/e19110581
[85]	D. Koutsoyiannis, Stochastics of Hydroclimatic Extremes - A Cool Look at Risk, Edition 3, 2023, ISBN: 978-618-85370-0-2,391 pages, Kallipos Open Academic Editions, Athens. https://doi.org/10.57713/kallipos-1
[86]	Climate Explorer, Dutch Royal Netherlands Meteorological Institute (KNMI), http://climexp.knmi.nl/(accessed on 25 January 2024).
[87]	Scripps CO₂ Program, Sampling Station Records, Available online, https://scrippsco2.ucsd.edu/data/atmospheric_co2/sampling_stations.html (accessed on 15 November 2023).
[88]	J. Ahn, M. Headly, M. Wahlen, E. J. Brook, P. A. Mayewski, K. C. Taylor, CO₂ diffusion in polar ice: observations from naturally formed CO₂ spikes in the Siple Dome (Antarctica) ice core, J. Glaciol., 54 (2008), 685–695. https://doi.org/10.3189/002214308786570764 doi: 10.3189/002214308786570764
[89]	C. W. Granger, Investigating causal relations by econometric models and cross-spectral methods, Econometrica, 37 (1969), 424–438. https://doi.org/10.2307/1912791 doi: 10.2307/1912791
[90]	C. W. Granger, Testing for causality: A personal viewpoint, J. Econ. Dyn. Control, 2 (1980), 329–352. https://doi.org/10.1016/0165-1889(80)90069-X doi: 10.1016/0165-1889(80)90069-X
[91]	J. Pearl, Causal inference in statistics: An overview, Stat. Surv., 3 (2009), 96–146. https://doi.org/10.1214/09-SS057 doi: 10.1214/09-SS057
[92]	J. Pearl, M. Glymour, N.P. Jewell, Causal Inference in Statistics: A Primer, Wiley, Chichester, UK, 2016.
[93]	A. Hannart, J. Pearl, F. E. L. Otto, P. Naveau, M. Ghil, Causal counterfactual theory for the attribution of weather and climate-related events, Bull. Amer. Met. Soc., 97 (2016), 99–110. https://doi.org/10.1175/BAMS-D-14-00034.1 doi: 10.1175/BAMS-D-14-00034.1
[94]	S. Arrhenius, On the influence of carbonic acid in the air upon the temperature of the ground, Lond. Edinb. Dublin Philos. Mag. J. Sci., 41 (1896), 237–276. https://doi.org/10.1080/14786449608620846 doi: 10.1080/14786449608620846
[95]	A. Prokoph, G. A. Shields, J. Veizer, Compilation and time-series analysis of a marine carbonate δ¹⁸O, δ¹³C, ⁸⁷Sr/⁸⁶Sr and δ³⁴S database through Earth history, Earth Sci. Rev., 87 (2008), 113–133. https://doi.org/10.1016/j.earscirev.2007.12.003 doi: 10.1016/j.earscirev.2007.12.003
[96]	J. Park, A re-evaluation of the coherence between global-average atmospheric CO₂ and temperatures at interannual time scales, Geophys. Res. Lett., 36 (2009), L22704. https://doi.org/10.1029/2009GL040975 doi: 10.1029/2009GL040975
[97]	L. Åsbrink, Revisiting causality using stochastics on atmospheric temperature and CO₂ concentration, Proc. R. Soc. A, 479 (2023), 20220529. https://doi.org/10.1098/rspa.2022.0529 doi: 10.1098/rspa.2022.0529
[98]	R. Weiss, Carbon dioxide in water and seawater: The solubility of a non-ideal gas, Mar. Chem. 2 (1974), 203–215. https://doi.org/10.1016/0304-4203(74)90015-2 doi: 10.1016/0304-4203(74)90015-2

mbe-21-07-287 Appendix.docx

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)