Heavy metals effects on life traits of juveniles of <em>Procambarus clarkii</em>

Paloma Alcorlo; Irene Lozano; Angel Baltanás; Paloma Alcorlo; Irene Lozano; Angel Baltanás

doi:10.3934/environsci.2019.3.147

AIMS Environmental Science

2019, Volume 6, Issue 3: 147-166. doi: 10.3934/environsci.2019.3.147

Previous Article Next Article

Research article Topical Sections

Heavy metals effects on life traits of juveniles of Procambarus clarkii

1.
Department of Ecology, Faculty of Sciences, Universidad Autónoma de Madrid, C/Darwin nº2, 28049 Madrid, Spain
2.
Golder Associates Global Ibérica S.L.U. Paseo de la Castellana, 140-3º Izda. Edificio Lima, 28046, Madrid, Spain

Received: 28 January 2019 Accepted: 15 May 2019 Published: 31 May 2019

An incubation experiment of juvenile crayfish (Procambarus clarkii) following a three- level treatment design approach was performed to assess the effect of different heavy metal concentrations on their life history traits (lifespan, growth, moult and feeding activity). The aims were to: (1) address the response of the life traits; (2) check for the correlation between heavy metal concentrations in crayfish whole bodies with the ones of the experimental solutions; (3) analyse the variation of crayfish carbon and nitrogen stable isotopes signatures grown under these treatments. Treatments were: control or absence of pollutants (C); low level contamination (L) similar to those found in the water of the Guadiamar River (SW, Spain) one year after the Aznalcóllar mine accident, and high level contamination (H) maximum concentrations of metals measured in the water of the river after the spill. The study concludes that the H treatment produced lethal effects on juveniles of crayfish, whereas those undergoing the L treatment showed less marked effects. Crayfish’s juveniles grown in L treatment seemed able to regulate and manage this range of pollution while maintaining their biological traits. Juvenile’s capacity to bioaccumulate toxic substances also changes with the nature of the particular metals. The reduction in lifespan was mainly influenced by Cu, Zn and As. 13 C of C and juveniles from L treatment had similar values but different from those individuals of H treatment, reflecting the isotopic signature of the food source used (liver), and were also influenced by the concentration of Cu and As.

Keywords:

Citation: Paloma Alcorlo, Irene Lozano, Angel Baltanás. Heavy metals effects on life traits of juveniles of Procambarus clarkii[J]. AIMS Environmental Science, 2019, 6(3): 147-166. doi: 10.3934/environsci.2019.3.147

Related Papers:

[1]	Maoxiang Wang, Fenglan Hu, Meng Xu, Zhipeng Qiu . Keep, break and breakout in food chains with two and three species. Mathematical Biosciences and Engineering, 2021, 18(1): 817-836. doi: 10.3934/mbe.2021043
[2]	Hao Wang, Yang Kuang . Alternative models for cyclic lemming dynamics. Mathematical Biosciences and Engineering, 2007, 4(1): 85-99. doi: 10.3934/mbe.2007.4.85
[3]	Minus van Baalen, Atsushi Yamauchi . Competition for resources may reinforce the evolution of altruism in spatially structured populations. Mathematical Biosciences and Engineering, 2019, 16(5): 3694-3717. doi: 10.3934/mbe.2019183
[4]	Xiyan Yang, Zihao Wang, Yahao Wu, Tianshou Zhou, Jiajun Zhang . Kinetic characteristics of transcriptional bursting in a complex gene model with cyclic promoter structure. Mathematical Biosciences and Engineering, 2022, 19(4): 3313-3336. doi: 10.3934/mbe.2022153
[5]	A. Swierniak, M. Krzeslak, D. Borys, M. Kimmel . The role of interventions in the cancer evolution–an evolutionary games approach. Mathematical Biosciences and Engineering, 2019, 16(1): 265-291. doi: 10.3934/mbe.2019014
[6]	Vaibhava Srivastava, Eric M. Takyi, Rana D. Parshad . The effect of "fear" on two species competition. Mathematical Biosciences and Engineering, 2023, 20(5): 8814-8855. doi: 10.3934/mbe.2023388
[7]	Natalia L. Komarova . Mathematical modeling of cyclic treatments of chronic myeloid leukemia. Mathematical Biosciences and Engineering, 2011, 8(2): 289-306. doi: 10.3934/mbe.2011.8.289
[8]	Fabio Augusto Milner, Ruijun Zhao . A deterministic model of schistosomiasis with spatial structure. Mathematical Biosciences and Engineering, 2008, 5(3): 505-522. doi: 10.3934/mbe.2008.5.505
[9]	Ali Moussaoui, Vitaly Volpert . The influence of immune cells on the existence of virus quasi-species. Mathematical Biosciences and Engineering, 2023, 20(9): 15942-15961. doi: 10.3934/mbe.2023710
[10]	Jim M. Cushing . A Darwinian version of the Leslie logistic model for age-structured populations. Mathematical Biosciences and Engineering, 2025, 22(6): 1263-1279. doi: 10.3934/mbe.2025047

Abstract

1. Introduction

One of interactions between three types of populations is known as non-transitive, cyclic competition or a rock-paper-scissors relationship, that is, type $A$ is stronger than type $B$ , and type $B$ is superior to type $C$ , which in turn is better than $A$ . We can find several examples of this kind of interaction in nature; the most well-known are bacteria (e.g., ^[1,2,3,4]) and lizards (e.g., ^[5,6,7]).

Barreto et al. ^[8] proposed phenotypic and genotypic cyclic competition models to clarify the fact that three throat morphs in male lizards, known as orange, blue, and yellow, are maintained with a rock-paper-scissors relationship. Both models have the same equilibrium, but the latter gives a wider stability region than the former. Barreto et al. ^[8] analyzed their model by fixing all payoffs except two elements, which are given by variable parameters. However, such a case is difficult to manage because an internal equilibrium depends on these parameters. Then we consider an alternative setting as an internal equilibrium with equal frequencies for all phenotypes independent of parameters, namely, we use symmetric conditions of parameters to obtain global insights into the dynamics in this simplified system for the convenience of analysis.

In this paper, we re-examine stability conditions under the following situations: symmetric parameters for the phenotypic model, cyclic allele dominance rule, and spatial structure. We obtain the following results: (ⅰ) Cyclic allele dominance rule in a genotypic model gives a wider stable region of internal equilibrium than the allele dominance rule observed in lizards. (ⅱ) Spatial structure drastically changes the dynamical behavior, especially when all three phenotypes coexist in almost all the parameter spaces when both competition and dispersal occur locally.

In the next section, we review the phenotypic model and its corresponding genotypic model proposed by ^[8] and then restrict these models with two parameters to generate an internal equilibrium independent of the parameters. In section 3, we derive local stability conditions for each model. We consider cyclic allele dominance rule in section 4. In section 5, we investigate the effects of spatial structure on the stability of the internal equilibrium.

2. The phenotypic model and its corresponding genotypic model

First, we consider a phenotypic model of cyclic competition, or a rock-paper-scissors game, following ^[8]. There are three phenotypes $O, B,$ and $Y$ . $O$ has a better strategy than $B$ , and $B$ is superior to $Y$ , which is, in turn, surpasses $O$ . When we use a payoff matrix in , this relation can be expressed by $M_{OY} < M_{OO} < M_{OB}, M_{BO} < M_{BB} < M_{BY},$ and $M_{YB} < M_{YY} < M_{YO}$ , where $M_{IJ}$ is a payoff of focal individual $I \in \{O, B, Y\}$ against an opponent $J \in \{O, B, Y\}$ .

Table 1. Payoff matrix of rock-paper-scissors game.

$M_{IJ}$ is a payoff of focal individual

$I \in \{O, B, Y\}$ against an opponent

$J \in \{O, B, Y\}$ . The following magnitude relations hold:

$M_{OY} < M_{OO} < M_{OB}, M_{BO} < M_{BB} < M_{BY}, M_{YB} < M_{YY} < M_{YO}$ .

	Individual $J$
Individual $I$	$O$	$B$	$Y$
$O$	$M_{OO}$	$M_{OB}$	$M_{OY}$
$B$	$M_{BO}$	$M_{BB}$	$M_{BY}$
$Y$	$M_{YO}$	$M_{YB}$	$M_{YY}$

| Show Table

DownLoad: CSV

Using these payoffs obtained by the competition between two phenotypes, the dynamics of phenotypic model are defined as

$\begin{eqnarray} p_{O, n+1} & = & \frac{W_O}{W} p_{O, n}, \end{eqnarray}$

(2.1)

$\begin{eqnarray} p_{B, n+1} & = & \frac{W_B}{W} p_{B, n}, \end{eqnarray}$

(2.2)

$\begin{eqnarray} p_{Y, n+1} & = & \frac{W_Y}{W} p_{Y, n}, \end{eqnarray}$

(2.3)

with $p_{O, n}, p_{B, n},$ and $p_{Y, n}$ as the fractions of phenotypes $O, B,$ and $Y$ , respectively, at generation $n$ . Here $W_O, W_B,$ and $W_Y$ are the fitnesses of phenotypes $O, B,$ and $Y$ , respectively, and $W$ is the average fitness of the population:

$\begin{eqnarray} W_O & = & M_{OO} p_{O, n}+M_{OB} p_{B, n}+M_{OY} p_{Y, n}, \end{eqnarray}$

(2.4)

$\begin{eqnarray} W_B & = & M_{BO} p_{O, n}+M_{BB} p_{B, n}+M_{BY} p_{Y, n}, \end{eqnarray}$

(2.5)

$\begin{eqnarray} W_Y & = & M_{YO} p_{O, n}+M_{YB} p_{B, n}+M_{YY} p_{Y, n}, \end{eqnarray}$

(2.6)

$\begin{eqnarray} W & = & W_O p_{O, n} + W_B p_{B, n} + W_B p_{Y, n}. \end{eqnarray}$

(2.7)

This is a standard model of the rock-paper-scissors game (e.g., ^[9,10]).

At equilibrium we have

$\begin{equation} W = W_O = W_B = W_Y \end{equation}$

(2.8)

by eqs.(2.1)–(2.3). A unique internal equilibrium solution for the above simultaneous equations (2.8) with $p_O + p_B + p_Y = 1$ is

$\begin{eqnarray} p_O & = & \frac{M_1}{M_1 + M_2 + M_3}, \end{eqnarray}$

(2.9)

$\begin{eqnarray} p_B & = & \frac{M_2}{M_1 + M_2 + M_3}, \end{eqnarray}$

(2.10)

$\begin{eqnarray} p_Y & = & \frac{M_3}{M_1 + M_2 + M_3}, \end{eqnarray}$

(2.11)

where

$\begin{eqnarray} M_1 & = & M_{OB}(M_{BY}-M_{YY}) + M_{BB}(M_{YY}-M_{OY}) + M_{YB}(M_{OY}-M_{BY}), \end{eqnarray}$

(2.12)

$\begin{eqnarray} M_2 & = & M_{OY}(M_{BO}-M_{YO}) + M_{BY}(M_{YO}-M_{OO}) + M_{YY}(M_{OO}-M_{BO}), \end{eqnarray}$

(2.13)

$\begin{eqnarray} M_3 & = & M_{OO}(M_{BB}-M_{YB}) + M_{BO}(M_{YB}-M_{OB}) + M_{YO}(M_{OB}-M_{BB}). \end{eqnarray}$

(2.14)

We have another three equilibria $(1, 0, 0), (0, 1, 0),$ and $(0, 0, 1)$ as the boundary equilibria $(\hat{p}_O, \hat{p}_B, \hat{p}_Y)$ .

Next we move to a genotypic model corresponding to the above phenotypic model by the genotypic allele dominance rule in which an allele $o$ dominates over two others, $y$ and $b$ , and an allele $y$ dominates $b$ . In other words, genotypes $oo, oy,$ and $ob$ are phenotype $O$ , genotypes $yy$ and $yb$ are phenotype $Y$ , and genotype $bb$ is phenotype $B$ (Table 2).

Table 2. Allele dominance rule (Ⅰ). An allele

$o$ is dominant to other alleles

$y$ and

$b$ , then three genotypes

$oo$ ,

$oy$ , and

$ob$ correspond to a phenotype

$O$ . An allele

$y$ is dominant to an allele

$b$ , then two genotypes

$yy$ and

$yb$ correspond to a phenotype

$Y$ . An allele

$b$ is recessive to other alleles, then only one genotype

$bb$ corresponds to a phenotype

$B$ .

Genotype		Phenotype
$oo, oy, ob$	$\to$	$O$
$yy, yb$	$\to$	$Y$
$bb$	$\to$	$B$

| Show Table

DownLoad: CSV

Then phenotypic frequencies can be expressed by the genotypic frequencies $g_{ij, n} \ (i, j \in \{o, y, b\})$ at generation $n$ as follows:

$\begin{eqnarray} p_{O, n} & = & g_{oo, n} + g_{oy, n} + g_{ob, n}, \end{eqnarray}$

(2.15)

$\begin{eqnarray} p_{Y, n} & = & g_{yy, n} + g_{yb, n}, \end{eqnarray}$

(2.16)

$\begin{eqnarray} p_{B, n} & = & g_{bb, n}. \end{eqnarray}$

(2.17)

Therefore the fitness of each genotype becomes

$\begin{eqnarray} W_O & = & W_{oo} = W_{oy} = W_{ob}, \end{eqnarray}$

(2.18)

$\begin{eqnarray} W_Y & = & W_{yy} = W_{yb}, \end{eqnarray}$

(2.19)

$\begin{eqnarray} W_B & = & W_{bb}. \end{eqnarray}$

(2.20)

Considering the above relationships of fitnesses and frequencies, first, the genotypic frequencies at generation $n$ change:

$\begin{equation} \tilde{g}_{ij, n} = \frac{W_{ij}}{W} g_{ij, n} \qquad (i, j \in \{o, y, b\}). \end{equation}$

(2.21)

Second, the allele frequencies $\tilde{f}_i \ (i \in \{o, y, b\})$ at generation $n$ are

$\begin{eqnarray} \tilde{f}_{o, n} & = & \tilde{g}_{oo, n} + \frac{1}{2} \tilde{g}_{oy, n} + \frac{1}{2} \tilde{g}_{ob, n}, \end{eqnarray}$

(2.22)

$\begin{eqnarray} \tilde{f}_{y, n} & = & \tilde{g}_{yy, n} + \frac{1}{2} \tilde{g}_{yb, n} + \frac{1}{2} \tilde{g}_{oy, n}, \end{eqnarray}$

(2.23)

$\begin{eqnarray} \tilde{f}_{b, n} & = & \tilde{g}_{bb, n} + \frac{1}{2} \tilde{g}_{yb, n} + \frac{1}{2} \tilde{g}_{ob, n}. \end{eqnarray}$

(2.24)

Third, assuming random mating gives the genotypic frequencies at generation $n+1$ , $g_{ij, n+1} \ (i, j \in \{o, y, b\})$ as

$\begin{eqnarray} g_{ij, n+1} & = & 2\tilde{f}_{i, n} \tilde{f}_{j, n} \quad \mbox{if $i \ne j$}, \end{eqnarray}$

(2.25)

$\begin{eqnarray} g_{ii, n+1} & = & \tilde{f}_{i, n}^2, \end{eqnarray}$

(2.26)

then we can describe the above relations and dynamics (2.15)–(2.26) with only variables representing the genotypic frequencies $g_{ij, n} (i, j \in \{0, y, b\})$ as

$\begin{eqnarray} g_{oo, n+1} & = & \left(\frac{W_O}{W}\right)^2\left(g_{oo, n} + \frac{1}{2} g_{oy, n} + \frac{1}{2} g_{ob, n}\right)^2, \end{eqnarray}$

(2.27)

$\begin{eqnarray} g_{oy, n+1} & = & 2\frac{W_O}{W}\left(g_{oo, n} + \frac{1}{2} g_{oy, n} + \frac{1}{2} g_{ob, n}\right) \\ && \times \left\{\frac{W_Y}{W}\left(g_{yy, n} + \frac{1}{2} g_{yb, n}\right) + \frac{1}{2} \frac{W_{O}g_{oy, n}}{W}\right\}, \end{eqnarray}$

(2.28)

$\begin{eqnarray} g_{ob, n+1} & = & 2\frac{W_O}{W}\left(g_{oo, n} + \frac{1}{2} g_{oy, n} + \frac{1}{2} g_{ob, n}\right) \\ && \times \left(\frac{W_{B}g_{bb, n}}{W} + \frac{1}{2} \frac{W_{Y}g_{yb, n}}{W} + \frac{1}{2} \frac{W_{O}g_{ob, n}}{W}\right), \end{eqnarray}$

(2.29)

$\begin{eqnarray} g_{yy, n+1} & = & \left\{\frac{W_Y}{W}\left(g_{yy, n} + \frac{1}{2} g_{yb, n}\right) + \frac{1}{2} \frac{W_{O}g_{oy, n}}{W}\right\}^2, \end{eqnarray}$

(2.30)

$\begin{eqnarray} g_{yb, n+1} & = & 2\left\{\frac{W_Y}{W}\left(g_{yy, n} + \frac{1}{2} g_{yb, n}\right) + \frac{1}{2} \frac{W_{O}g_{oy, n}}{W}\right\} \\ && \times \left(\frac{W_{B}g_{bb, n}}{W} + \frac{1}{2} \frac{W_{Y}g_{yb, n}}{W} + \frac{1}{2} \frac{W_{O}g_{ob, n}}{W}\right), \end{eqnarray}$

(2.31)

$\begin{eqnarray} g_{bb, n+1} & = & \left(\frac{W_{B}g_{bb, n}}{W} + \frac{1}{2} \frac{W_{Y}g_{yb, n}}{W} + \frac{1}{2} \frac{W_{O}g_{ob, n}}{W}\right)^2, \end{eqnarray}$

(2.32)

where

$\begin{eqnarray} W_O & = & M_{OO} (g_{oo, n} + g_{oy, n} + g_{ob, n})+M_{OB} g_{bb, n}+M_{OY} (g_{yy, n} + g_{yb, n}), \end{eqnarray}$

(2.33)

$\begin{eqnarray} W_B & = & M_{BO} (g_{oo, n} + g_{oy, n} + g_{ob, n})+M_{BB} g_{bb, n}+M_{BY} (g_{yy, n} + g_{yb, n}), \end{eqnarray}$

(2.34)

$\begin{eqnarray} W_Y & = & M_{YO} (g_{oo, n} + g_{oy, n} + g_{ob, n})+M_{YB} g_{bb, n}+M_{YY} (g_{yy, n} + g_{yb, n}), \end{eqnarray}$

(2.35)

$\begin{eqnarray} W & = & W_O(g_{oo, n} + g_{oy, n} + g_{ob, n}) + W_B g_{bb, n} + W_Y(g_{yy, n} + g_{yb, n}). \end{eqnarray}$

(2.36)

Using the above equations, we can obtain the following relations:

$\begin{eqnarray} g_{oo, n+1}+\frac{1}{2}g_{oy, n+1}+\frac{1}{2}g_{ob, n+1} & = & \frac{W_O}{W} \left(g_{oo, n}+\frac{1}{2}g_{oy, n}+\frac{1}{2}g_{ob, n}\right), \end{eqnarray}$

(2.37)

$\begin{eqnarray} g_{yy, n+1}+\frac{1}{2}g_{yb, n+1}+\frac{1}{2}g_{oy, n+1} & = & \frac{W_Y}{W} \left(g_{yy, n}+\frac{1}{2}g_{yb, n}\right)+ \frac{1}{2}\frac{W_O}{W}g_{oy, n}, \end{eqnarray}$

(2.38)

$\begin{eqnarray} g_{bb, n+1}+\frac{1}{2}g_{yb, n+1}+\frac{1}{2}g_{ob, n+1} & = & \frac{W_B}{W} g_{bb, n}+\frac{1}{2}\frac{W_Y}{W} g_{yb, n}+ \frac{1}{2}\frac{W_O}{W} g_{ob, n}. \end{eqnarray}$

(2.39)

At equilibrium, by eq.(2.37), we have

$\begin{equation} W = W_O. \end{equation}$

(2.40)

By eq.(2.38) and eq.(2.40), we have

$\begin{equation} W = W_Y. \end{equation}$

(2.41)

By eqs.(2.39)–(2.41), we have

$\begin{equation*} W = W_B. \end{equation*}$

Therefore, because we also have the same relationships (2.8) for genetypic model, we have the same phenotypic equilibrium (2.9)–(2.11) with eqs.(2.12)–(2.14) as the corresponding phenotypic model.

3. Stability condition for equilibrium $(1/3, 1/3, 1/3)$

The payoff matrix includes nine payoffs and is a little complicated. Following ^[8], we reduce it to two values. We fix the values of the payoffs at 1 for the competitions between the same phenotypes, but we also adopt the symmetric cases --- namely, we give a larger payoff $\alpha > 1$ to a stronger competitor and a smaller one $\beta < 1$ to a weaker competitor (), which results in the same equilibrium fractions of $1/3$ for all three phenotypes ^[9].

Table 3. Payoff matrix with only two parameters. We obtain the same equilibrium fractions of

$1/3$ for all three phenotypes. Parameter ranges are given by

$0 < \beta < 1 < \alpha$ .

	Opponent Individual
Focal Individual	$O$	$B$	$Y$
$O$	$1$	$\alpha$	$\beta$
$B$	$\beta$	$1$	$\alpha$
$Y$	$\alpha$	$\beta$	$1$

| Show Table

DownLoad: CSV

By using this reduced payoff matrix, eqs.(2.1)–(2.7) become

$\begin{eqnarray} p_{O, n+1} & = & \frac{(p_{O, n}+\alpha p_{B, n}+\beta p_{Y, n})p_{O, n}}{(p_{O, n}+\alpha p_{B, n}+\beta p_{Y, n})p_{O, n} + (\beta p_{O, n}+ p_{B, n}+\alpha p_{Y, n})p_{B, n} +(\alpha p_{O, n}+\beta p_{B, n}+ p_{Y, n})p_{Y, n}}, \end{eqnarray}$

(3.1)

$\begin{eqnarray} p_{B, n+1} & = & \frac{(\beta p_{O, n}+ p_{B, n}+\alpha p_{Y, n})p_{B, n}}{(p_{O, n}+\alpha p_{B, n}+\beta p_{Y, n})p_{O, n} + (\beta p_{O, n}+ p_{B, n}+\alpha p_{Y, n})p_{B, n} +(\alpha p_{O, n}+\beta p_{B, n}+ p_{Y, n})p_{Y, n}}, \end{eqnarray}$

(3.2)

$\begin{eqnarray} p_{Y, n+1} & = & \frac{(\alpha p_{O, n}+\beta p_{B, n}+ p_{Y, n})p_{Y, n}}{(p_{O, n}+\alpha p_{B, n}+\beta p_{Y, n})p_{O, n} + (\beta p_{O, n}+ p_{B, n}+\alpha p_{Y, n})p_{B, n} +(\alpha p_{O, n}+\beta p_{B, n}+ p_{Y, n})p_{Y, n}}. \end{eqnarray}$

(3.3)

By eqs.(2.12)–(2.14),

$\begin{equation*} M_1 = M_2 = M_3 = \alpha^2+\beta^2 -(\alpha+\beta+\alpha\beta) + 1, \end{equation*}$

so that an internal equilibrium (2.9)–(2.11) becomes

$\begin{equation} p_O = p_Y = p_B = \frac{1}{3}. \end{equation}$

(3.4)

The linearized system of eqs.(3.1)–(3.3) about an internal equilibrium (3.4) gives the following Jacobian matrix:

$\begin{equation*} \frac{1}{3(1+\alpha+\beta)} \begin{pmatrix} 4+\alpha+\beta & -2+\alpha-2\beta & -2-2\alpha+\beta \\ -2-2\alpha+\beta & 4+\alpha+\beta & -2+\alpha-2\beta \\ -2+\alpha-2\beta & -2-2\alpha+\beta & 4+\alpha+\beta \end{pmatrix}. \end{equation*}$

Then the characteristic equation of the linearized system becomes

$\begin{equation*} \lambda(\lambda^2 + a_1 \lambda + a_2) = 0, \end{equation*}$

where

$\begin{eqnarray} a_1 & = & -\frac{4+\alpha+\beta}{1+\alpha+\beta}, \end{eqnarray}$

(3.5)

$\begin{eqnarray} a_2 & = & \frac{4+2(\alpha+\beta)+\alpha^2+\beta^2}{(1+\alpha+\beta)^2}. \end{eqnarray}$

(3.6)

Jury conditions or Schur-Cohn criteria (e.g., ^[11]) for second-degree characteristic equations $\lambda^2 + a_1 \lambda + a_2 = 0$ are known as

$\begin{equation} |a_1| \lt 1 + a_2 \lt 2. \end{equation}$

(3.7)

Because $a_1 < 0$ from eq.(3.5), the left inequality of eq.(3.7) becomes $-a_1 < 1 + a_2$ , that is, $a_1 + a_2 + 1 > 0$ , which holds by the calculation using eqs.(3.5)–(3.6):

$\begin{equation*} a_1 + a_2 + 1 = \frac{(1-\alpha)^2+(1-\beta)^2+(\alpha-\beta)^2}{2(1+\alpha+\beta)^2} \gt 0. \end{equation*}$

The right inequality of eq.(3.7) $1 + a_2 < 2$ , that is, $a_2 -1 < 0,$ gives

$\begin{equation} \alpha\beta \gt 1. \end{equation}$

(3.8)

Notice that it is symmetric with respect to $\alpha$ and $\beta$ ; namely, eq.(3.8) does not change when $\alpha$ and $\beta$ are exchanged.

Indeed, it is known that eq.(3.8) becomes a globally asymptotic stable condition, which can be checked using the Lyapunov function ^[9]. If eq.(3.8) does not hold, then we can observe the trajectory of the heteroclinic cycle between three vertices in the triangular space for three phenotypic frequencies ^[9] similar to the three-species Lotka-Volterra cyclic competition model ^[12] (See Figure 1(a) and (b)).

Figure 1. Trajectories of phenotypic frequencies. (a)-(c) for eqs.(3.1)-(3.3), (d)-(f) for eqs.(2.27)-(2.36) with Table 3.

$\beta$ is fixed to

$0.3$ .

$\alpha = 2$ for (a) and (d),

$\alpha = 3$ for (b) and (e),

$\alpha = 4$ for (c) and (f). Figure 2(a) tells us that internal equilibrium is unstable for (a), (b) and (d), but stable for (c), (e) and (f). Colorbars indicate the time steps of the dynamics.

DownLoad: Full-Size Img PowerPoint

Figure 2. Stable and unstable region on parameter spaces of the equilibrium point (3.4) for (a) allele dominance rule (Ⅰ) and (b) allele dominance rule (Ⅱ).

$\alpha$ and

$\beta$ indicate payoffs for strong and weak competitors, respectively. Green, red and yellow regions indicate stable for both phenotypic and genotypic models, stable for genotypic model but unstable for phenotypic one and unstable for both phenotypic and genotypic models, respectively. Allele dominance rule (Ⅱ) has larger stability region than (Ⅰ).

DownLoad: Full-Size Img PowerPoint

On the other hand, for a genotypic model, we get the following unique internal equilibrium by eqs.(2.27)–(2.36):

$\begin{eqnarray} g_{oo} & = & \frac{\left(\sqrt{3}-\sqrt{2}\right)^2}{2}, \end{eqnarray}$

(3.9)

$\begin{eqnarray} g_{oy} & = & \left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{2}-1\right), \end{eqnarray}$

(3.10)

$\begin{eqnarray} g_{ob} & = & \frac{2}{3}\left(\sqrt{3}-\sqrt{2}\right), \end{eqnarray}$

(3.11)

$\begin{eqnarray} g_{yy} & = & \frac{1}{3}\left(\sqrt{2}-1\right)^2, \end{eqnarray}$

(3.12)

$\begin{eqnarray} g_{yb} & = & \frac{2}{3}\left(\sqrt{2}-1\right), \end{eqnarray}$

(3.13)

$\begin{eqnarray} g_{bb} & = & \frac{1}{3}. \end{eqnarray}$

(3.14)

Of course,

$\begin{eqnarray*} p_O & = & g_{oo} + g_{oy} + g_{ob} = \frac{1}{3}, \\ p_Y & = & g_{yy} + g_{yb} = \frac{1}{3}, \\ P_B & = & g_{bb} = \frac{1}{3}. \end{eqnarray*}$

We can say that an internal equilibrium does not depend on the parameters $\alpha$ nor $\beta$ on either the phenotyic or genotypic models.

The characteristic equation of the linearized system of eqs.(2.27)–(2.32) with eqs.(2.33)–(2.36) about an internal equilibrium (3.9)–(3.14) becomes

$\begin{equation*} \lambda^4(\lambda^2 + a_1 \lambda + a_2) = 0, \end{equation*}$

where

$\begin{eqnarray} a_1 & = & -\frac{2\sqrt{3}}{3(1+\alpha+\beta)} \left\{\left(2\sqrt{2}+2-\sqrt{3}\right)+\left(2\sqrt{3}-\sqrt{2}-1\right)(\alpha+\beta)\right\}, \end{eqnarray}$

(3.15)

$\begin{eqnarray} a_2 & = & \frac{1}{(1+\alpha+\beta)^2} \left[8\sqrt{2}-11+4\sqrt{3}\left(\sqrt{2}-2\right) \right. \\ &+& \left\{2\sqrt{2}\left(4+\sqrt{3}\right)-\left(6\sqrt{3}+5\right)\right\}(\alpha^2+\beta^2) \\ &+& 2\left\{4+3\sqrt{3}-\sqrt{2}\left(4+\sqrt{3}\right)\right\}(\alpha+\beta) \\ &-& \left. 2\left\{2\sqrt{2}\left(2+\sqrt{3}\right)-\left(7+2\sqrt{3}\right)\right\}\alpha\beta \right]. \end{eqnarray}$

(3.16)

Noticing that $a_1 < 0$ from eq.(3.15), the left inequality of eq.(3.7) holds because, using eqs.(3.15)–(3.16),

$\begin{eqnarray*} && a_1 + a_2 + 1 \\ & = & \frac{4\sqrt{3}\left(\sqrt{3}-\sqrt{2}\right)\left(\sqrt{2}-1\right)}{3(1+\alpha+\beta)^2} \left\{(1-\alpha)^2+(1-\beta)^2+(\alpha-\beta)^2\right\} \gt 0. \end{eqnarray*}$

The right inequality of eq.(3.7) $a_2 -1 < 0$ becomes

$\begin{equation} \sqrt{2}\left(\sqrt{3}+1\right)\left(\sqrt{3}+\sqrt{2}\right)(\alpha\beta-1) + \alpha(\alpha-1)+\beta(\beta-1) \gt 0, \end{equation}$

(3.17)

which is a quadratic inequation on $\alpha$ or $\beta$ and is also symmetric with respect to $\alpha$ and $\beta$ .

Figure 2(a) shows the local stability region obtained by eq.(3.8) and eq.(3.17). We can observe their dynamics by numerical simulations (Figure 1).

4. Cyclic allele dominance rule expands the stability region of the internal equilibrium

Common side-blotched lizards have an allele dominance rule (Ⅰ), that is, an allele $o$ for an orange throat is the most dominant over others, an allele $y$ for a yellow one is intermediate, and an allele $b$ for a blue one is the most recessive (). In other words, this allele dominance rule has three kinds of alleles: most dominant, intermediate, and most recessive. We refer to these as $o$ , $y$ , and $b$ , respectively.

Although perhaps we have not yet discovered it in realistic genetic systems, we can theoretically consider another allele dominance rule (Ⅱ), that is, all alleles having intermediate dominance: $o$ dominant over $y$ , $y$ over $b$ , and $b$ over $o$ (). This rule could be called as a "cyclic allele dominance rule." Similarly, in this rule, we name these alleles $o$ , $y$ , and $b$ , and any allele can be replaced by another one.

Table 4. Allele dominance rule (Ⅱ). An allele

$o$ is dominant to an allele

$y$ , then two genotypes

$oo$ and

$oy$ correspond to a phenotype

$O$ . An allele

$y$ is dominant to an allele

$b$ , then two genotypes

$yy$ and

$yb$ correspond to a phenotype

$Y$ . An allele

$b$ is dominant to an allele

$o$ , then two genotypes

$bb$ and

$ob$ correspond to a phenotype

$B$ .

Genotype		Phenotype
$oo, oy$	$\to$	$O$
$yy, yb$	$\to$	$Y$
$bb, ob$	$\to$	$B$

| Show Table

DownLoad: CSV

A calculation for allele dominance rule (Ⅱ) similar to that for allele dominance rule (Ⅰ) reveals that the boundary of local stability obtained from the Jury condition (3.7) is determined by

$\begin{equation} 8(\alpha\beta-1)+\alpha(\alpha-1)+\beta(\beta-1) = 0, \end{equation}$

(4.1)

which only differs from the first calculation in the smaller coefficient of $\alpha\beta-1$ compared to eq.(3.17). Figure 2(b) shows the local stability region by allele dominance rule (Ⅱ) using eq.(4.1), which clearly has a larger stability region than rule (Ⅰ).

From a theoretical point of view, we should also consider the cases with the same allele dominance rules but with different cyclic competitive strengths. Then another possible combination exists, that is, with $\alpha < 1$ and $\beta > 1$ . However, this combination gives the same stability regions as by the symmetry of the boundary equation (3.17) on the parameters $\alpha$ and $\beta$ ; eq.(3.17) does not change when $\alpha$ and $\beta$ are exchanged. On the other hand, it is trivial that no qualitative difference exists between the two cases: with $\alpha > 1$ and $\beta < 1$ and with $\alpha < 1$ and $\beta > 1$ .

5. Spatial simulation

Barreto et al. ^[8] showed the same stabilizing effect in a genetic system of lizards as the previous section of this paper. Here we introduce a spatial structure into a phenotypic model and consider its effects on population dynamics.

The phenotypic model includes two processes: competition and reproduction. When we add a spatial structure into this model, we restrict the spatial range, both for the opponents against a focal individual and for the dispersion of offspring by reproduction. When we set such a spatial restriction as a whole or as a neighborhood, we consider four distinct cases: (a) global competition and global dispersion, (b) global competition and local dispersion, (c) local competition and global dispersion, and (d) local competition and local dispersion.

The Monte Carlo simulation procedures for the above four cases use the following algorithm (See the C program code in the Electronic Supplementary Materials):

(ⅰ) We prepare a two-dimensional square lattice space, each-side with a size of 100, so that the total number of sites on a whole lattice becomes $N = 100\times 100 = 10000$ . We use a periodic boundary condition. The states $O$ , $Y$ , and $B$ are randomly distributed according to the initial fractions: $(p_O, p_Y, p_B) = (0.3, 0.3, 0.4)$ , respectively.

(ⅱ) The first process is competition, which determines the payoffs after games against randomly chosen opponents. All individuals experience this process. The payoffs for all the sites $i \ ( = 1, \ldots, N)$ in a whole population with size $N$ are determined in order; the site $(\ell, m)$ with $x$ -coordinate $\ell \ (\ell = 1, \ldots, 100)$ and $y$ -coordinate $m \ (m = 1, \ldots, 100)$ on the two-dimensional square lattice is numbered $(\ell-1)\times 100 + m$ . An individual $I \in\{O, B, Y\}$ is included on the site $i$ . Another site $j$ is randomly chosen in a whole population in the cases of (a) and (b) or, in the cases of (c) and (d), in the nearest neighboring four individuals. An individual $J \in\{O, B, Y\}$ is included on the site $j$ . Two individuals $I$ and $J$ compete and an individual $I$ gets a payoff by a payoff matrix (Table 3). All the payoffs are transferred to their offspring, and they are gathered in one offspring pool in the cases of (a) and (c) or, in the cases of (b) and (d), in local offspring pools. Here, we use lower cases for the sites and corresponding upper cases for the states of the individuals on those sites.

(ⅲ) The second process is dispersion, which gives positions of offspring selected from the offspring pool. The model is discrete-generation: all individuals die, and the states at the next time steps for all the sites $i \ ( = 1, \ldots, N)$ in a whole population with size $N$ are replaced in order. For each site, an individual is randomly chosen from a whole pool for (a) and (c) or from each local pool for (b) and (d); this choice is made in proportion to the relative payoff against total payoffs in a whole population in the cases of (a) and (c) or in the nearest neighboring four individuals in the cases of (b) and (d). In other words, this random selection depends on $\frac{W_O}{W}, \frac{W_Y}{W}$ , and $\frac{W_B}{W}$ . After the previous process (ⅱ), all the individuals have their own payoffs. In the cases of (a) and (c), we can then calculate $W, W_O, W_Y$ , and $W_B$ as the total payoffs for, respectively, a whole population, phenotype $O$ , phenotype $Y$ , and phenotype $B$ . In the cases of (b) and (d), we can obtain $W, W_O, W_Y$ , and $W_B$ as the total payoffs for, respectively, a local population, phenotype $O$ in the local population, phenotype $Y$ in the local population, and phenotype $B$ in the local population. Here a local population is restricted to individuals on five sites: the focal site and its nearest neighboring four sites.

(ⅳ) The above procedures (ⅱ) and (ⅲ) are repeated for $10000 \ ( = 100 \times 100)$ time steps for each parameter combination of $\alpha = 1, 1.05, 1.10, \ldots, 5$ and $\beta = 0, 0.05, 0.10, \ldots, 1$ . The repetition of the simulations is 100 for each parameter combination, and we record the fraction of the coexistence of the three phenotypes.

We show the results of the Monte Carlo simulation by the above algorithm in Figure 3. Case (a) completely coincides with Figure 2(a). Case (b) gives a wider coexistence region due to local dispersion. Case (c) has the same result as case (a) because the global dispersal produces the same effect with random choices of opponents from the population. Case (d) shows coexistence in almost all the simulations for the entire parameter space. Therefore, we can conclude that locally limited interactions between individuals strongly promote the coexistence of all phenotypes.

Figure 3. Phase diagram for coexistence and non-coexistence. Each parameter

$\alpha$ or

$\beta$ is set for every 0.05 increment. Initial values are given as

$(p_O, p_B, p_Y) = (0.3, 0.3, 0.4)$ . Simulation runs are repeated for 100 times for each parameter set and their fractions are shown by continuous gradation between green and yellow. Green indicates coexistence until the end of simulations. Yellow corresponds to non-coexistence in which one phenotype goes extinct, and then an inferior phenotype disappears and ultimately only one phenotype survives, due to finite size of the system. (Upper left panel) global competition and global dispersion; (Upper right panel) global competition and local dispersion; (Lower left panel) local competition and global dispersion; (Lower right panel) local competition and local dispersion.

DownLoad: Full-Size Img PowerPoint

In case (d), individuals with the same phenotypes tend to cluster and adopt collective behavior, so changes of state occur only on the boundaries between these clusters, causing a slower change of total population and stabilization (See Figure 4). Indeed ^[13] already reached such a conclusion from a continuous-time cyclic competition model that also calculated the average domain size or boundary length at a steady state. This spatial structure in the population is gradually produced by both local competition and local dispersion, and, in turn, also gives different results in competition and dispersion, either globally or locally.

Figure 4. Snapshot of spatial pattern by Monte Carlo simulation of phenotypic model on two-dimensional square lattice space after some time. Both competition and dispersion occur locally. We can observe that several dozens of same phenotypic individuals are clustering. (Right panel) The whole spatial pattern with

$1000 \times 1000$ lattice sites. (Left panel) The upper left part of spatial pattern with

$100 \times 100$ lattice sites. Colors indicate

$O, B, Y$ as orange, blue, yellow, respectively.

$\alpha = 2.0$ and

$\beta = 0.3$ .

DownLoad: Full-Size Img PowerPoint

6. Discussion

Similar to ^[8], we show the difference between the phenotypic model and its corresponding genotypic model, but we can also give an explicit condition using a simplified system. In addition, we investigate other allele dominance rules to clarify their effects on the stability of internal equilibrium compared to the allele dominance rule observed in lizards. Unfortunately, however, we cannot specify the mechanism by which the genotypic model stabilizes the system more than the phenotypic model. To discover it, we need to find models other than cyclic competition in which the genotypic models give different dynamic behaviors than the corresponding phenotypic models.

A recent proposal by ^[14] may have important implications for future work in this area. As they point out, the present three-strategy payoff matrix can be built up as the sum of nine independent Fourier components, $\mathbf{g}(1), \ldots, \mathbf{g}(9)$ , with orthogonality and normalization:

$\begin{eqnarray*} \begin{pmatrix} 1 & \alpha & \beta \\ \beta & 1 & \alpha \\ \alpha & \beta & 1 \end{pmatrix} & = & (1+\alpha+\beta) \cdot \frac{1}{3} \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} \\ &+& \frac{\sqrt{2}}{2}(2-\alpha-\beta) \cdot \frac{1}{\sqrt{18}} \begin{pmatrix} 2 & -1 & -1 \\ -1 & 2 & -1 \\ -1 & -1 & 2 \end{pmatrix} \\ &+& \frac{\sqrt{6}}{2}(\beta-\alpha) \cdot \frac{1}{\sqrt{6}} \begin{pmatrix} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0 \end{pmatrix} \\ & = & (1+\alpha+\beta) \mathbf{g}(1) + \frac{\sqrt{2}}{2}(2-\alpha-\beta) \mathbf{g}(8) + \frac{\sqrt{6}}{2}(\beta-\alpha) \mathbf{g}(9), \end{eqnarray*}$

where we have $\mathbf{g}(1), \mathbf{g}(8)$ , and $\mathbf{g}(9)$ as irrelevant constant terms, a coordination (Potts) component that equally favors the formation of one of the homogeneous states, and a cyclic (rock-paper-scissors) component, respectively. This formulation may offer more detailed insights into dynamical behaviors if we examine cases with various combinations of coefficients corresponding to the strengths of these components.

The intuitive reason local competition and local dispersion both stabilize an internal equilibrium is that several dozens of identical phenotypic individuals gather and adopt collective behavior, restricting phenotypic changes on the boundary. To investigate their stabilization mechanisms in detail, we should further rely on other kinds of analysis, especially spatial pattern formation, which may play an important role in stabilizing the dynamics. So far, the concept of "vortex" has been proposed to characterize the spatial pattern ^[15,16,17], and it has also been shown that clockwise or anti-clockwise rotating spiral patterns with characteristic sizes emerge that yield finite size effects by cyclic competition models or rock-paper-scissors games in physics ^{[18,19,20,21]} and evolutionary games ^[22,23]. More precisely, the coexistence of three strategies transforms into one of the homogeneous states after a suitable relaxation time if the characteristic length of patterns exceeds a threshold value comparable to the system size. Here we use a small lattice space size $(100 \times 100)$ for , but we should further investigate these finite size effects using a larger lattice space, such as $1000 \times 1000$ , as in Figure 4.

In this paper, we only show whether three phenotypes can coexist or not in the long run using a Monte Carlo simulation. However, we can expect an internal equilibrium to be asymptotically stable for the case of coexistence in larger Monte Carlo systems than in smaller ones (Figure 5).

Figure 5. Time series of frequencies by Monte Carlo simulation of phenotypic model on two-dimensional square lattice space. Both competition and dispersion occur locally. We can confirm that fluctuations are due to the effect of finite size of lattice sites. The numbers of lattice sites are (a)

$1000 \times 1000$ , (b)

$100 \times 100$ , (c)

$10 \times 10$ . All phenotypes survived after 2000 Monte Carlo steps in (a) and (b), but in (c)

$B$ went extinct around 70 generations by chance, and then

$O$ died out almost inevitably (but not definitely because of finite system size) because

$Y$ is stronger than

$O$ . Colors indicate

$O, B, Y$ as orange, blue, yellow, respectively. Initial frequencies are

$(p_O, p_B, p_Y) = (0.3, 0.3, 0.4)$ .

$\alpha = 2.0$ and

$\beta = 0.3$ .

DownLoad: Full-Size Img PowerPoint

Here we can only show the condition of the locally asymptotic stability of a genotypic model. We expect that condition to be replaced by global stability, but we leave it as a future problem.

In addition, we would like to examine a genotypic model in a lattice space and clarify whether it becomes more stable than a phenotypic model and to what degree its stability can be evaluated by, for example, the return time of perturbation to an internal equilibrium.

Throughout this paper, we use only symmetric conditions of parameters to obtain global insights into the dynamics, then we should study more complicated models with symmetry-breaking interactions, including the parameter setting of ^[8], in future research.

Acknowledgements

I sincerely appreciate Prof. Akira Sasaki for giving me useful comments and encouraging me. I would also like to thank three anonymous reviewers whose comments were helpful in revising the manuscript.

Conflict of interest

The author declares no conflict of interest.

References

[1]	Sabater S (2008) Alterations of the global water cycle and their effects on river structure, function and services. Freshw Rev 1: 75–88. doi: 10.1608/FRJ-1.1.5
[2]	Prat N, Toja J, Solá C, et al. (1999) Effect of dumping and cleaning activities on the aquatic ecosystems of the Guadiamar River following a toxic flood. Sci Tot Environ 242: 231–248. doi: 10.1016/S0048-9697(99)00393-9
[3]	Grimalt JO, Ferrern M, Macpherson E (1999) The mine tailing accident in Aznalcóllar. Sci Tot Environ 242: 3–11. doi: 10.1016/S0048-9697(99)00372-1
[4]	Arenas JM, Carrascal JF (2004) Situación medioambiental del Corredor Verde del Guadiamar 6 años después del vertido de Aznalcóllar. Ecosystemas 13: 69–78.
[5]	Habsburgo-Lorena AS (1979) Preset situation of exotics species of crayfish introduced into Spanish continental waters. Freshw Crayfish 4: 175–84.
[6]	Gutiérrez-Yurrita PJ (1997). El papel ecológico del cangrejo rojo (Procambarus clarkii) en los ecosistemas acuáticos del Parque Nacional de Doñana. Una perspectiva ecofisiológica y bioenergética. PhThesis. Dept. de Ecología. Universidad Autónoma de Madrid.
[7]	Gherardi F (2006) Crayfish Invading Europe, the Case Study of Procambarus clarkii. Mar Freshw Behav Phy 39: 175–191. doi: 10.1080/10236240600869702
[8]	Maranhão P, Marques JC, Madeira V (1995) Copper concentrations in soft tissues of the red swamp crayfish Procambarus clarkii (Girard, 1852), after exposure to a range of dissolved copper concentrations. Freshw Crayfish 10: 282–286.
[9]	Naqvi SM, Flagge CT (1990). Chronic effects of arsenic on American red crayfish, Procambarus clarkii, exposed to Monosodium Methanearsonate (MSMA) Herbicide. B Environ Contam Tox 45: 101–106. doi: 10.1007/BF01701835
[10]	Antón A, Serrano T, Angulo E, et al. (2000) The use of two species of crayfish as environmental quality sentinels: the relationship between heavy metal content, cell and tissue biomarkers and physico-chemical characteristics of the environment. Sci Tot Environ 247: 239–251. doi: 10.1016/S0048-9697(99)00493-3
[11]	Alcorlo P, Otero M, Crehuet M et al. (2006) The use of the red swamp crayfish (Procambarus clarkii, Girard) as indicator of the bioavailability of heavy metals in environmental monitoring in the River Guadiamar (SW, Spain). Sci Tot Environ 366: 380–390. doi: 10.1016/j.scitotenv.2006.02.023
[12]	Martín-Díaz ML, Tuberty SR, McKenney C, et al. (2006) The use of bioaccumulation, biomarkers and histopathology diseases in Procambarus clarkii to establish bioavailability of Cd and Zn after a mining spill. Environ Monit Assess 116: 169–184. doi: 10.1007/s10661-006-7234-0
[13]	Vioque-Fernández A, Alves de Almeida E, Ballesteros J, et al. (2007). Doñana National Park survey using crayfish (Procambarus clarkii) as bioindicator: esterase inhibition and pollutant levels. Toxicol Lett 168: 260–268. doi: 10.1016/j.toxlet.2006.10.023
[14]	Vioque-Fernández A, Alves de Almeida, E, López-Barea J (2009) Assessment of Doñana National Park contamination in Procambarus clarkii: Integration of conventional biomarkers and proteomic approaches. Sci Tot Environ 407: 1784–1797. doi: 10.1016/j.scitotenv.2008.11.051
[15]	Faria M, Huertas D, Soto DX, et al. (2010) Contaminant accumulation and multi-biomarker responses in field collected zebra mussels (Dreissena polymorpha) and crayfish (Procambarus clarkii), to evaluate toxicological effects of industrial hazardous dumps in the Ebro river (NE Spain). Chemosphere 78: 232–240. doi: 10.1016/j.chemosphere.2009.11.003
[16]	Suárez-Serrano A, Alcaraz C, Ibáńez C, et al. (2010) Procambarus clarkii as a bioindicator of heavy metal pollution sources in the lower Ebro River and Delta. Ecotox Environ Saf 73: 280–286. doi: 10.1016/j.ecoenv.2009.11.001
[17]	Geiger G, Alcorlo P, Baltanás A, et al. (2005) Impact of an introduced Crustacean on the trophic webs of Mediterranean wetlands. Biol Invasions 7: 49–73. doi: 10.1007/s10530-004-9635-8
[18]	Tablado Z, Tella JL, Sánchez-Zapata JA, et al. (2010) The paradox of the long-term positive effects of a North American crayfish on a European Community of predators. Conserv Biol 24: 1230–1238. doi: 10.1111/j.1523-1739.2010.01483.x
[19]	Rodríguez EM, Medesani DA, Fingerman M (2007) Endocrine disruption in crustaceans due to pollutants: a review. Comp Biochem Phys A 146: 661–671. doi: 10.1016/j.cbpa.2006.04.030
[20]	Kouba A, Buric M, Kozák P (2010) Bioaccumulation and effects of heavy metals in crayfish: a review. Wat Air Soil Pollut 211: 5–16. doi: 10.1007/s11270-009-0273-8
[21]	Peterson BJ, Howarth RW, Garritt RH (1985) Multiple stable isotopes used to trace the flow of organic matter in estuarine food webs. Science 227: 1361–1363. doi: 10.1126/science.227.4692.1361
[22]	Peterson BJ, Fry B (1987) Stable isotopes in ecosystem studies. Annu Rev Ecol and Syst 18: 293–320. doi: 10.1146/annurev.es.18.110187.001453
[23]	Lajtha K, Michener RH (1994) Introduction, In: Lajtha K, Michener RH, Stable isotopes in ecology and environmental science, Eds., London, UK: Blackwell Scientific Publication.
[24]	Hershey AE, Peterson BJ (1996) Stream food webs. In: Hauer FR, Lamberti GA, Methods in stream ecology, Eds., San Diego, California: Academic Press, 511–529.
[25]	Gannes LZ, Río CM, Koch P (1998) Natural abundance variations in stable isotopes and their potential uses in animal physiological ecology. Comp Biochem Phys C 119: 725–737. doi: 10.1016/S1095-6433(98)01016-2
[26]	Inger R, Bearhop S (2008) Applications of stable isotope analysis to avian ecology. Ibis 150: 447–461. doi: 10.1111/j.1474-919X.2008.00839.x
[27]	Martínez del Rio C, Wolf N, Carleton SA, et al. (2009) Isotopic ecology ten years after a call for more laboratory experiments. Biol Rev Camb Philos 84: 91–111. doi: 10.1111/j.1469-185X.2008.00064.x
[28]	Deniro MJ, Epstein S (1978) Influence of diet on the distribution of carbon isotopes in animals. Geochim Cosmochim Ac 42: 495–506. doi: 10.1016/0016-7037(78)90199-0
[29]	Deniro MJ, Epstein S (1981) Influence of diet on the distribution of nitrogen isotopes in animals. Geochim Cosmochim Ac 45: 341–351. doi: 10.1016/0016-7037(81)90244-1
[30]	Reynolds JD (2002) Growth and Reproduction, In: D.M. Holdich Biology of Freshwater Crayfish Blackw, Eds, UK: Blackwell Science, 152–191.
[31]	Nyström P (2002) Ecology, In: Holdich DM. Biology of Freshwater Crayfish. Ed., UK: Blackwell Science, 192–224.
[32]	Martín G, Alcalá E, Burgos MD, et al. (2004) Efecto de la contaminación minera sobre el perifiton del río Guadiamar. Limnetica 23: 315–330.
[33]	Toja J (2008) Efecto del accidente minero en el perifiton del río Guadiamar. Las algas bentónicas como indicadoras de la calidad del agua. In: Redondo I, Montes C., Carrascal F, La restauración ecológica del río Guadiamar y el proyecto del Corredor Verde. La historia de un paisaje emergente, Eds, Consejería de Medio Ambiente. Junta de Andalucía: 205–220.
[34]	Chen WJ, Wu J, Malone RF (1995) Effects of temperature on mean molt interval, molting and mortality of Red Swamp Crawfish. Aquaculture 131: 205–217. doi: 10.1016/0044-8486(94)00327-K
[35]	Paglianti A, Gherardi F (2004) Combined effects of temperature and diet on growth and survival of young-of-year crayfish: a comparison between indigenous and invasive species. J Crus Biol 24: 140–148. doi: 10.1651/C-2374
[36]	Del Ramo J, Díaz-Mayans J, Torreblanca A, et al. (1987) Effects of temperature on the acute toxicity of heavy metals (Cr, Cd and Hg) to the freshwater crayfish Procambarus clarkii (Girard). B Environ Contam Tox 38: 736–741. doi: 10.1007/BF01616694
[37]	Zar JH (1999) Biostatistical Analysis, 4th Ed, New Jersey, USA.
[38]	Anderson MB, Reddy P, Preslan JE, et al. (1997a) Metal accumulation in crayfish, exposed to a petroleum-contaminated Bayou in Louisiana. Ecotox Environ Safe 37: 267–272.
[39]	Anderson MB, Preslan JE, Jolibois L, et al. (1997b) Bioaccumulation of lead nitrate in Red Swamp Crayfish (Procambarus clarkii). J Hazard Mater 54: 15–29.
[40]	Mirenda RJ (1986) Toxicity and accumulation of cadmium in the crayfish, Orconectes virilis(Hagen). Arch Environ Con Tox 15: 401–407. doi: 10.1007/BF01066407
[41]	Naqvi SM, Howell RD (1993) Toxicity of cadmium and lead to juvenile red swamp crayfish, Procambarus clarkii, and effects on fecundity of adults. B Environ Contam Tox 51: 303–308.
[42]	Martínez M, Torreblanca A, Del Ramo J, et al. (1994) Effects of sublethal exposure to lead on levels of energetic compounds in Procambarus clarkii (Girard, 1852). B Environ Contam Tox 52: 729–733. doi: 10.1007/BF00195495
[43]	Taylor RM, Watson GD, Alikhan MA (1995) Comparative sub-lethal and letal acute toxicity of copper to the freshwater crayfish, Cambarus robutus (Cambaridae, Decapoda, Crustacea) from an acidic metal-contaminated lake and a circumneutral uncontaminated stream. Water Resour 29: 401–408.
[44]	Rainbow PS, Amiard-Triquet C, Amiard JC, et al. (2000) Observations on the interaction of zinc and cadmium uptake rates in crustaceans (amphipods and crabs) from coastal sites in UK and France differentially enriched with trace metals. Aquat Toxicol 50: 189–204. doi: 10.1016/S0166-445X(99)00103-4
[45]	Bardeggia M, Alikhan MA (1991) The relationship between copper and nickel levels in the diet, and their uptake and accumulation by Cambarus bartoni (Fabricius) (Decapoda, Crustacea). Water Resour 25(10): 1187–1192
[46]	Kim SD (2003) The removal by crab shell of mixed heavy metal ions in aqueous solution. Bioresource Technol 87: 355–357. doi: 10.1016/S0960-8524(02)00259-6
[47]	Torreblanca A, Díaz-Mayans J, Del Ramo J (1987) Oxygen uptake and gill morphological alterations in Procambarus clarkii (Girard) after sublethal exposure to lead. Comp Biochem Phys C 86: 219–224. doi: 10.1016/0742-8413(87)90167-8
[48]	Vosloo A, Van Aardt WJ, Mienie LJ (2002) Sublethal effects of copper on the freshwater crab Potamonautes warreni. Comp Biochem Phys C 133: 695–702. doi: 10.1016/S1095-6433(02)00214-3
[49]	Rowe CL, Hopkins WA, Zehnder C, et al. (2000) Metabolic costs incurred by crayfish (Procambarus acutus) in a trace element-polluted habitat: further evidence of similar responses among diverse taxonomic groups. Comp Biochem Phys C, 129: 275–283.
[50]	López-López S, Nolasco H, Vega-Villasante F (2003) Characterization of digestive gland esterase-lipase activity of juvenile redclaw crayfish Cherax quadricarinatus. Comp Biochem Phys C 135: 337–347.
[51]	Sherba M, Dunham DW, Harvey HH (2000) Sublethal copper toxicity and food response in the freshwter crayfish Cambarus bartonii (Cambridae, Decapoda, Crustacea). Ecotox Environ Saf 46: 329–333. doi: 10.1006/eesa.1999.1910
[52]	Weis JS, Cristini A, Rao KK (1992) Effects of pollutants on molting and regeneration in Crustacea. Amer Zool 32: 495–500. doi: 10.1093/icb/32.3.495
[53]	Chen JC, Lin CH (2001) Toxicity of copper sulfate for survival, growth, molting and feeding of juveniles of the tiger shrimp, Penaeus monodon. Aquaculture 192: 55–65. doi: 10.1016/S0044-8486(00)00442-7
[54]	Carmona-Osalde C, Rodríguez-Serna M, Olvera-Novoa MA et al (2004) Gonadal development, spawning, growth and survival of the crayfish Procambarus llamasi at three different water temperatures. Aquaculture 232: 305–316. doi: 10.1016/S0044-8486(03)00527-1
[55]	Zanotto FP, Wheatly MG (2003) Calcium balance in crustaceans: nutritional aspects of physiological regulation. Comp Biochem Phys A 133: 645–660.
[56]	Stinson MD, Eaton DL (1983) Concentrations of Lead, Cadmium, Mercury, and Koper in the Crayfish (Pacifastacus leniusculus) Obtained from a Lake Receiving Urban Runoff. Arch Environ Con Tox 12: 693–700. doi: 10.1007/BF01060753
[57]	Depledge MH, Forbes TL, Forbes VE (1993) Evaluation of cadmium, copper, zinc and iron concentrations and tissue distributions in the benthic crab, Dorippe granulata (De Haan, 1841) from Tolo Harbour, Hong Kong. Environ Pollut 81: 15–19. doi: 10.1016/0269-7491(93)90023-H
[58]	Rainbow PS (1995). Physiology, Physicochemistry and metal uptake: a crustacean perspective. Marine Pollut Bull 31: 55–59. doi: 10.1016/0025-326X(95)00005-8
[59]	Zia S, Alikhan MA (1989) Copper uptake and regulation in a copper-tolerant decapod Cambarus bartoni (Fabricius) (Decapoda, Crustacea). B Environ Contam Tox 42: 103–110.
[60]	Rainbow PS (1997). Ecophysiology of trace metal uptake in Crustaceans. Estuar Coast Shelf S 44: 169–175. doi: 10.1006/ecss.1996.0208
[61]	Alcorlo P, Geiger W, Otero, M (2008) Reproductive biology and life cycle of the invasive crayfish Procambarus clarkii (Crustacea: Decapoda) in diverse aquatic habitats of South-Western Spain: Implications for population control. Fund Appl Limnol 173(3): 197–212.
[62]	Allison G, Laurenson LJB, Pistone G, et al. (2000) Effects of dietary copper on the Australian Freshwater Crayfish Cherax destructor. Ecotox Environ Safe 46: 117–123. doi: 10.1006/eesa.1999.1863
[63]	Pastor A, Medina J, Del Ramo J, et al. (1988) Determination of lead in treated crayfish Procambarus clarkii: Accumulation in different tissues. B Environ Contam Tox 41: 412–418. doi: 10.1007/BF01688887
[64]	Rincón-Leon F, Zurera-Cosano G, Pozo-Lora R (1988) Lead and cadmium concentrations in Red Crayfish (Procambarus clarkii, G.) in the Guadalquivir River Marshes (Spain). Arch Environ Con Tox 17: 251–256.
[65]	Del Ramo J, Pastor A, Torreblanca A, et al. (1989) Cadmium-Blinding proteins in midgut gland of freshwater crayfish Procambarus clarkii. B Environ Contam Tox 42: 241–246. doi: 10.1007/BF01699406
[66]	Naqvi SM, Glagge CT, Hawkins RL (1990) Arsenic uptake and depuration by red crayfish, Procambarus clarkii, exposed to various concentrations of Monosodium Methanearsonate (MSMA) Herbicide. B Environ Contam Tox 45: 94–100. doi: 10.1007/BF01701834
[67]	Junger M, Planas D (1994) Quantitative use of stable carbon isotope analysis to determine the trophic base of invertebrate communities in a boreal forest lotic system. Can J Fish Aquat Sci 51: 52–61. doi: 10.1139/f94-007
[68]	Quinn MR, Feng X, Folt CL, et al. (2003) Analyzing trophic transfer of metals in stream food webs using nitrogen isotopes. Sci Tot Environ, 17: 73–89.
[69]	Larsson P, Holmqvist N, Stenroth P, et al. (2007) Heavy Metals and Stable Isotopes in a Benthic Omnivore in a Trophic Gradient of Lakes. Environ Sci Technol 41(17): 5973–5979.
[70]	Alcorlo P, Baltanás A (2013) The trophic ecology of the red swamp crayfish (Procambarus clarkii) in Mediterranean aquatic ecosystems: a stable isotope study. Limnetica 32(1): 121–138.
[71]	Power M, Klein G., Guiguer KRRA, et al. (2002) Mercury accumulation in the fish community of a sub-arctic lake in relation to trophic position and carbon sources. J Appl Ecol 39: 819–830. doi: 10.1046/j.1365-2664.2002.00758.x
[72]	Watanabe K, Monaghan MT, Takemon Y, et al. (2008) Biodilution of heavy metals in a stream macroinvertebrate food web: Evidence from stable isotope analysis. Sci Tot Environ 394: 57–67. doi: 10.1016/j.scitotenv.2008.01.006

This article has been cited by:

1.	A. Szolnoki, B. F. de Oliveira, D. Bazeia, Pattern formations driven by cyclic interactions: A brief review of recent developments, 2020, 131, 1286-4854, 68001, 10.1209/0295-5075/131/68001
2.	Sourav Kumar Sasmal, Yasuhiro Takeuchi, Editorial: Mathematical Modeling to Solve the Problems in Life Sciences, 2020, 17, 1551-0018, 2967, 10.3934/mbe.2020167

Reader Comments

Your name:*

Email:*
© 2019 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)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Environmental Science

1.2 2.7

Metrics

Article views(6041) PDF downloads(1465) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

Figures and Tables

Figures(4) / Tables(5)

AIMS Environmental Science

Heavy metals effects on life traits of juveniles of Procambarus clarkii

Related Papers:

Abstract

1. Introduction

2. The phenotypic model and its corresponding genotypic model

3. Stability condition for equilibrium $(1/3, 1/3, 1/3)$

4. Cyclic allele dominance rule expands the stability region of the internal equilibrium

5. Spatial simulation

6. Discussion

Acknowledgements

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Catalog

AIMS Environmental Science

Heavy metals effects on life traits of juveniles of Procambarus clarkii

Related Papers:

Abstract

1. Introduction

2. The phenotypic model and its corresponding genotypic model

3. Stability condition for equilibrium (1/3,1/3,1/3) (1/3, 1/3, 1/3)

4. Cyclic allele dominance rule expands the stability region of the internal equilibrium

5. Spatial simulation

6. Discussion

Acknowledgements

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Figures and Tables

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

3. Stability condition for equilibrium $(1/3, 1/3, 1/3)$