Some Contributions to the Class of Branching Processes with Several Mating and Reproduction Strategies

Abstract

This work deals with mathematical modeling of dynamical systems. We consider a class of two-sex branching processes with several mating and reproduction strategies. We provide some probabilistic and statistical contributions. We deduce general expressions for the probability generating functions underlying the probability model, we derive some properties concerning the behavior of the states of the process and we determine estimates for the offspring mean vectors governing the reproduction phase. Furthermore, we extend the two-sex model considering immigration of female and male individuals from external populations. The results are illustrated through simulated examples. The investigated two-sex models are of particular interest to mathematically describe the population dynamics of biological species with a single reproductive episode before dying (semalparous species).

1. Introduction

In this work, we continue the research line about the class of two-sex branching processes with several mating and reproductive strategies introduced in [1]. Branching processes are usually used as mathematical models to describe the population dynamics of biological species, see, e.g., [2,3,4]. In particular, a fairly rich literature has emerged concerning discrete-time two-sex branching processes, see the surveys by [5,6], and discussions therein. Most of these stochastic models assume that all the couples female-male have identical reproductive behavior (they produce new female and male individuals according to the same offspring probability distribution) and also that mating and reproduction depend on the number of progenitor couples in the population, see, e.g., [7,8,9,10,11]. In many biological species, due to environmental factors, reproduction occurs in a non-predictable environment where both phases, mating and reproduction, usually are influenced by the current numbers of females and males in the population, see, e.g., [12]. In order to describe the probabilistic evolution of such species, branching processes had not been sufficiently investigated. To this purpose, in [1], a class of two-sex branching models was introduced. Several results about such a class of models have been derived in [13,14]. We continue this research line providing new probabilistic and statistical contributions.

The paper is organized as follows. In Section 2, we mathematically describe the probability model and we derive some theoretical contributions. In Section 3, we extend the probability model and the previous contributions incorporating immigration of females and males from external populations. We include illustrative examples. In Section 4, we present the concluding remarks and some questions for research.

2. Probability Model

In [1], on a probability space $(\Omega ,\mathcal{A},P)$, we introduced a two-sex branching process ${\left\{X_n\right\}}_{n=0}^{\infty }$, $X_n=(F_n,M_n)$ representing the number of female and male individuals at generation n. The probability model assumes $n_m\ge 1$ and $n_r\ge 1$ mating and reproduction strategies, respectively. It is described as follows, where $\mathbb{N}$ and ${\mathbb{N}}_{+}$ denote the non-negative and positive integers, respectively:

1.

Mating phase is represented by a sequence of $n_m$ two arguments integer-valued functions ${\left\{L_l\right\}}_{l\in {\mathbb{N}}_m}$, ${\mathbb{N}}_m:=\{1,\dots ,n_m\}$. Each $L_l$ is assumed to be non-decreasing and such that $L_l(f,m)\le fm,f,m\in \mathbb{N}$. At generation n, according to the l-th mating strategy, $L_l(F_n,M_n)$ couples female-male are formed.

2.

Reproduction phase is modeled by a sequence of $n_r$ probability distributions (offspring distributions) ${\left\{P_h\right\}}_{h\in {\mathbb{N}}_r}$, ${\mathbb{N}}_r:=\{1,\dots ,n_r\}$, $P_h:={\left\{p_{(f,m)}^h\right\}}_{(f,m)\in S_h}$, $S_h\subseteq {\mathbb{N}}^2$, $p_{(f,m)}^h$ being the probability that a given couple produces exactly f females and m males when $P_h$ is the underlying reproductive strategy.

3.

In each generation, the mating and reproduction strategies are determined through functions ${\phi }_m$ and ${\phi }_r$, both defined on ${\mathbb{N}}^2$ and taking values on ${\mathbb{N}}_m$ and ${\mathbb{N}}_r$, respectively.

We start with $X_0=(F_0,M_0)\in {\mathbb{N}}_{+}^2$. Then, given that at generation n, $X_n=x\in {\mathbb{N}}^2$, we obtain that $L_{{\phi }_m\left(x\right)}$ and $P_{{\phi }_r\left(x\right)}$ are the mating and reproductive strategies, respectively. Hence,

$$X_{n+1}=(F_{n+1},M_{n+1}):=\sum _{i=1}^{L_{{\phi }_m\left(x\right)}\left(x\right)}(F_{n,i}^{{\phi }_r\left(x\right)},M_{n,i}^{{\phi }_r\left(x\right)}),n\in \mathbb{N}$$

(1)

with $F_{n,i}^{{\phi }_r\left(x\right)}$ and $M_{n,i}^{{\phi }_r\left(x\right)}$ denoting, respectively, the number of female and male individuals originated by the i-th couple at generation n. For each $h\in {\mathbb{N}}_r$, independently of n, the random vectors $(F_{n,i}^h,M_{n,i}^h)$, $i=1,\dots ,L_{{\phi }_m\left(x\right)}\left(x\right)$, defined on $(\Omega ,\mathcal{A},P)$, are assumed to be i.i.d. with $P_h$ being the offspring distribution,

$$P(F_{n,1}^h=f,M_{n,1}^h=m)=p_{(f,m)}^h,(f,m)\in S_h$$

Given $x_0,\dots ,x_n,x_{n+1}\in {\mathbb{N}}^2$, by considering that independently of the generation n, the random vectors $(F_{n,i}^h,M_{n,i}^h)$ are i.i.d., it is derived from (1),

$$\begin{array}{cc}\hfill P(X_{n+1}=x_{n+1}\mid X_0=x_0,\dots ,X_n=x_n)& =P\left(\sum _{i=1}^{L_{{\phi }_m\left(x_n\right)}\left(x_n\right)}(F_{n,i}^{{\phi }_r\left(x_n\right)},M_{n,i}^{{\phi }_r\left(x_n\right)})=x_{n+1}\right)\hfill \\ \hfill & =P(X_{n+1}=x_{n+1}\mid X_n=x_n)\hfill \end{array}$$

Note that, the corresponding transition probabilities are independent of the generation n considered. In fact, for $x,z\in {\mathbb{N}}^2$,

$$P(X_{n+1}=z\mid X_n=x)=P\left(\sum _{i=1}^{L_{{\phi }_m\left(x\right)}\left(x\right)}(F_{n,i}^{{\phi }_r\left(x\right)},M_{n,i}^{{\phi }_r\left(x\right)})=z\right)=\sum _{z_1,\cdots ,z_d\in {\Delta }_z}\prod _{i=1}^d{p}_{z_i}^{{\phi }_r\left(x\right)}$$

where $d=L_{{\phi }_m\left(x\right)}\left(x\right)$ and ${\Delta }_z:=\{z_1,\dots ,z_d\in {\mathbb{N}}^2:{\displaystyle \sum _{i=1}^d}{z}_i=z\}$. Hence, ${\left\{X_n\right\}}_{n=0}^{\infty }$ is a homogeneous Markov chain with state space ${\mathbb{N}}^2$. Clearly, if for some n, $X_n=(0,0)$ then $X_{n+j}=(0,0)$, $j\ge 1$. Thus, $(0,0)$ is an absorbent state.

Remark 1.

Two-sex Model (1) is particularly appropriate to mathematically describe the population dynamics of semalparous species, namely, biological species with a single reproductive episode before dying. Functions $L_l$, ${\phi }_m$, and ${\phi }_r$, should be flexible enough in order to fit the main features of the species we pretend to describe. Usually, such functions will depend of biological/ethological parameters of interest in the demographic dynamics of the species.

Let $f_h(s,t):=E\left[s^{F_{0,1}^h}{t}^{M_{0,1}^h}\right]$ and $g_n(s,t):=E\left[s^{F_n}{t}^{M_n}\right],s,t\in [0,1]$, be the probability generating functions (p.g.f.) of $(F_{0,1}^h,M_{0,1}^h)$ and $(F_n,M_n)$, $h\in {\mathbb{N}}_r$, $n\in \mathbb{N}$, respectively. Clearly, $g_0(s,t)=s^{F_0}{t}^{M_0}$. Next result determines the general expression for $g_n,n\in {\mathbb{N}}_{+}$.

Proposition 1.

For $n\in \mathbb{N}$,

$$g_{n+1}(s,t)=E\left[{\left(f_{{\phi }_r\left(X_n\right)}(s,t)\right)}^{L_{{\phi }_m\left(X_n\right)}\left(X_n\right)}\right],s,t\in [0,1]$$

Proof. 

Given $n\in \mathbb{N}$,

$$\begin{array}{cc}\hfill g_{n+1}(s,t)& =E\left[s^{F_{n+1}}{t}^{M_{n+1}}\right]=E\left[E[s^{F_{n+1}}{t}^{M_{n+1}}\mid X_n]\right]\hfill \\ \hfill & =\sum _{x\in {\mathbb{N}}^2}E\left[s^{\sum _{i=1}^{L_{{\phi }_m\left(x\right)}\left(x\right)}{F}_{n,i}^{{\phi }_r\left(x\right)}}{t}^{\sum _{i=1}^{L_{{\phi }_m\left(x\right)}\left(x\right)}{M}_{n,i}^{{\phi }_r\left(x\right)}}\right]P(X_n=x)\hfill \\ \hfill & =\sum _{x\in {\mathbb{N}}^2}{\left(E\left[s^{F_{0,1}^{{\phi }_r\left(x\right)}}{t}^{M_{0,1}^{{\phi }_r\left(x\right)}}\right]\right)}^{L_{{\phi }_m\left(x\right)}\left(x\right)}P(X_n=x)\hfill \\ \hfill & =\sum _{x\in {\mathbb{N}}^2}{\left(f_{{\phi }_r\left(x\right)}(s,t)\right)}^{L_{{\phi }_m\left(x\right)}\left(x\right)}P(X_n=x)\hfill \\ \hfill & =E\left[{\left(f_{{\phi }_r\left(X_n\right)}(s,t)\right)}^{L_{{\phi }_m\left(X_n\right)}\left(X_n\right)}\right],s,t\in [0,1]\hfill \end{array}$$

□

We now provide some properties about the behavior of the states of ${\left\{X_n\right\}}_{n=0}^{\infty }$. To this purpose, we assume that $L_l$, $l\in {\mathbb{N}}_m$, are superadditive functions, i.e., given $n\in {\mathbb{N}}_{+}$,

$$L_l\left(\sum _{i=1}^n{x}_i\right)\ge \sum _{i=1}^n{L}_l\left(x_i\right),x_i\in {\mathbb{N}}^2,l\in {\mathbb{N}}_m$$

(2)

Superadditivity is a classical and logical requirement in two-sex branching process literature.

Furthermore, for $x\in {\mathbb{N}}^2$, independently of n, let

$$C_x:=\{y\in {\mathbb{N}}^2:P(X_{n+m}=y\mid X_n=x)>0forsomem\ge 1\}$$

be the set of states which can be reached from x.

Proposition 2.

Assume $x_0\in {\mathbb{N}}_{+}^2$ such that $p_{x_0}^h>0$, $L_l\left(x_0\right)>1$, $h\in {\mathbb{N}}_r$, $l\in {\mathbb{N}}_m$. Given $x\in {\mathbb{N}}^2$:

(a) 

There exists $x^{\prime }\in {\mathbb{N}}^2$ with $L_{{\phi }_m\left(x^{\prime }\right)}\left(x^{\prime }\right)>L_{{\phi }_m\left(x\right)}\left(x\right)$ verifying that $x^{\prime }\in C_{x_0}$.

(b) 

If $p_{(0,0)}^h>0,h\in {\mathbb{N}}_r$, then $(0,0)\in C_x$.

Proof. 

(a)

Let us introduce the sequence ${\left\{x_n\right\}}_{n=0}^{\infty }$, where

$$x_{n+1}:=x_0{L}_{{\phi }_m\left(x_n\right)}\left(x_n\right),n\in \mathbb{N}$$

By using (2) and the fact that $L_l\left(x_0\right)>1$, $l\in {\mathbb{N}}_m$,

$$\begin{array}{cc}\hfill L_{{\phi }_m\left(x_{n+1}\right)}\left(x_{n+1}\right)& =L_{{\phi }_m\left(x_{n+1}\right)}\left(x_0{L}_{{\phi }_m\left(x_n\right)}\left(x_n\right)\right)\hfill \\ \hfill & =L_{{\phi }_m\left(x_{n+1}\right)}\left(\sum _{i=1}^{L_{{\phi }_m\left(x_n\right)}\left(x_n\right)}{x}_0\right)\hfill \\ \hfill & \ge \sum _{i=1}^{L_{{\phi }_m\left(x_n\right)}\left(x_n\right)}{L}_{{\phi }_m\left(x_{n+1}\right)}\left(x_0\right)\hfill \\ \hfill & =L_{{\phi }_m\left(x_n\right)}\left(x_n\right)L_{{\phi }_m\left(x_{n+1}\right)}\left(x_0\right)\hfill \\ \hfill & >L_{{\phi }_m\left(x_n\right)}\left(x_n\right)\hfill \end{array}$$

Hence ${\left\{L_{{\phi }_m\left(x_n\right)}\left(x_n\right)\right\}}_{n=0}^{\infty }\nearrow \infty$. Consequently, given $x\in {\mathbb{N}}^2$, there exists $x_n$ such that $L_{{\phi }_m\left(x_n\right)}\left(x_n\right)>L_{{\phi }_m\left(x\right)}\left(x\right)$. Thus, it is sufficient to check that $x_n\in C_{x_0}$. In fact, if for some $l\in {\mathbb{N}}_{+}$, $X_l=x_0$, then:

$$P(X_{l+n}=x_n\mid X_l=x_0)\ge \prod _{i=0}^{n-1}P(X_{l+i+1}=x_{i+1}\mid X_{l+i}=x_i)=\prod _{i=0}^{n-1}P(X_{l+1}=x_{i+1}\mid X_l=x_i)$$

(3)

Now, using that $x_{i+1}=x_0{L}_{{\phi }_m\left(x_i\right)}\left(x_i\right)$ and $p_{x_0}^h>0$, $h\in {\mathbb{N}}_r$,

$$P(X_{l+1}=x_{i+1}\mid X_l=x_i)=P\left(\sum _{j=1}^{L_{{\phi }_m\left(x_i\right)}\left(x_i\right)}(F_{l,j}^{{\phi }_r\left(x_i\right)},M_{l,j}^{{\phi }_r\left(x_i\right)})=x_{i+1}\right)\ge {\left(p_{x_0}^{{\phi }_r\left(x_i\right)}\right)}^{L_{{\phi }_m\left(x_i\right)}\left(x_i\right)}>0$$

Therefore, from (3),

$$P(X_{l+n}=x_n\mid X_l=x_0)\ge {\left(p_{x_0}^{{\phi }_r\left(x_i\right)}\right)}^{\sum _{i=0}^{n-1}{L}_{{\phi }_m\left(x_i\right)}\left(x_i\right)}>0$$

(b)

If for some $n\in \mathbb{N}$, $X_n=x$, then:

$$P(X_{n+1}=(0,0)\mid X_n=x)=P\left(\sum _{j=1}^{L_{{\phi }_m\left(x\right)}\left(x\right)}(F_{n,j}^{{\phi }_r\left(x\right)},M_{n,j}^{{\phi }_r\left(x\right)})=(0,0)\right)={\left(p_{(0,0)}^{{\phi }_r\left(x\right)}\right)}^{L_{{\phi }_m\left(x\right)}\left(x\right)}>0$$

□

Let ${\mu }^h:=({\mu }_1^h,{\mu }_2^h)$ and ${\Delta }^h:={({\sigma }_{ij}^h)}_{i,j=1,2}$ be, respectively, the mean vector and the covariance matrix of $(F_{0,1}^h,M_{0,1}^h)$, $h\in {\mathbb{N}}_r$,

$${\mu }_i^h:=\sum _{(k_1,k_2)\in S_h}{k}_i{p}_{(k_1,k_2)}^h,{\sigma }_{ij}^h:=\sum _{(k_1,k_2)\in S_h}(k_i-{\mu }_i^h)(k_j-{\mu }_j^h)p_{(k_1,k_2)}^h,i,j=1,2.$$

For $n\in \mathbb{N}$ and $x\in {\mathbb{N}}^2$, let us denote by ${\mu }_{n+1}^x$ and ${\Delta }_{n+1}^x$ the mean vector and the covariance matrix, respectively, of $X_{n+1}$ given that $X_n=x$. From Proposition 1, it can be verified that, independently of n:

$$E[s^{F_{n+1}}{t}^{M_{n+1}}\mid X_n=x]={\left(f_{{\phi }_r\left(x\right)}(s,t)\right)}^{L_{{\phi }_m\left(x\right)}\left(x\right)},s,t\in [0,1]$$

$${\mu }_{n+1}^x=L_{{\phi }_m\left(x\right)}\left(x\right){\mu }^{{\phi }_r\left(x\right)},{\Delta }_{n+1}^x=L_{{\phi }_m\left(x\right)}\left(x\right){\Delta }^{{\phi }_r\left(x\right)}$$

(4)

Next, we consider the estimation of ${\mu }^h,h\in {\mathbb{N}}_r$. To this end, we will assume that, for some $n\in {\mathbb{N}}_{+}$, we know the observations of the variables:

$$X_0,L_{{\phi }_m\left(X_k\right)}\left(X_k\right),X_{k+1},k=0,\dots ,n$$

(5)

For each $h\in {\mathbb{N}}_r$, let $T_h:=\{k\in \{0,\dots ,n\}:{\phi }_r\left(X_k\right)=h\}$, i.e., the set of generations (until generation n) where $P_h$ has been the reproductive strategy.

Proposition 3.

Given $h\in {\mathbb{N}}_r$ such that $T_h\ne \varnothing$ and $\sum _{k\in T_h}{L}_{{\phi }_m\left(X_k\right)}\left(X_k\right)>0$, a conditional moment-based estimator for ${\mu }^h$ using the data sample (5), is given by:

$$\widehat{{\mu }^h}={\left(\sum _{k\in T_h}{L}_{{\phi }_m\left(X_k\right)}\left(X_k\right)\right)}^{-1}\sum _{k\in T_h}{X}_{k+1}$$

(6)

Proof. 

From (4),

$$E[X_{k+1}\mid X_k]=L_{{\phi }_m\left(X_k\right)}\left(X_k\right){\mu }^{{\phi }_r\left(X_k\right)}a.s.$$

Hence, by using the moment estimation procedure, we can propose as estimate for ${\mu }^{{\phi }_r\left(X_k\right)}$, based on the observations of $L_{{\phi }_m\left(X_k\right)}\left(X_k\right)$ and $X_{k+1}$,

$$\overline{{\mu }_{\left(k\right)}^{{\phi }_r\left(X_k\right)}}:={\left(L_{{\phi }_m\left(X_k\right)}\left(X_k\right)\right)}^{-1}{X}_{k+1},k=0,\dots ,n$$

(7)

It is verified that,

$$E\left[\overline{{\mu }_{\left(k\right)}^{{\phi }_r\left(X_k\right)}}\mid L_{{\phi }_m\left(X_k\right)}\left(X_k\right)>0\right]={\mu }^{{\phi }_r\left(X_k\right)}$$

In fact,

$$\begin{array}{cc}\hfill E\left[\overline{{\mu }_{\left(k\right)}^{{\phi }_r\left(X_k\right)}}\mid L_{{\phi }_m\left(X_k\right)}\left(X_k\right)>0\right]& =\frac{\sum _{z\in {\mathbb{N}}_{+}}{z}^{-1}E[X_{k+1}\mid L_{{\phi }_m\left(X_k\right)}\left(X_k\right)=z]P(L_{{\phi }_m\left(X_k\right)}\left(X_k\right)=z)}{P(L_{{\phi }_m\left(X_k\right)}\left(X_k\right)>0)}\hfill \\ \hfill & =\frac{\sum _{z\in {\mathbb{N}}_{+}}{z}^{-1}E\left[\sum _{i=1}^z(F_{k,i}^{{\phi }_k\left(X_k\right)},M_{k,i}^{{\phi }_k\left(X_k\right)})\right]P(L_{{\phi }_m\left(X_k\right)}\left(X_k\right)=z)}{P(L_{{\phi }_m\left(X_k\right)}\left(X_k\right)>0)}\hfill \\ \hfill & =\frac{{\mu }^{{\phi }_r\left(X_k\right)}\sum _{z\in {\mathbb{N}}_{+}}P(L_{{\phi }_m\left(X_k\right)}\left(X_k\right)=z)}{P(L_{{\phi }_m\left(X_k\right)}\left(X_k\right)>0)}={\mu }^{{\phi }_r\left(X_k\right)}\hfill \end{array}$$

Thus, by considering (7), we propose as appropriate estimator for ${\mu }^h$:

$$\widehat{{\mu }^h}:=\sum _{k\in T_h}{\beta }_k^h\overline{{\mu }_{\left(k\right)}^h}$$

(8)

where $\sum _{k\in T_h}{\beta }_k^h=1$. Taking ${\beta }_k^h\propto L_{{\phi }_m\left(X_k\right)}\left(X_k\right)$, we deduce,

$${\beta }_k^h={\left(\sum _{k\in T_h}{L}_{{\phi }_m\left(X_k\right)}\left(X_k\right)\right)}^{-1}{L}_{{\phi }_m\left(X_k\right)}\left(X_k\right)$$

Consequently, from (7) and (8), we derive the Expression (6). □

Example 1.

Let us consider a two-sex model (1) where, given $(f,m)\in {\mathbb{N}}^2$,

1. 

Females and males form couples through the $n_m=2$ mating strategies:

$$L_1(f,m)=\lfloor K_{1}fmin\{1,m\}\rfloor ,L_2(f,m)=\lfloor K_{1}mmin\{f,1\}\rfloor$$

$\lfloor ·\rfloor$ denoting integer part and $K_1\in (0,1)$ representing the estimated proportion of individuals in the population which disappears due to environmental factors.

2. 

The couples produce new female and male individuals according to the $n_r=2$ reproductive strategies $P_h=\left\{p_{(f,m)}^h\right\},h=1,2$, where:

$$p_{(f,m)}^1:=P(F_{0,1}^1=f,M_{0,1}^1=m)=e^{-2.5}{(1.3)}^f{(1.2)}^m{(f!m!)}^{-1}$$

(9)

$$p_{(f,m)}^2:=P(F_{0,1}^2=f,M_{0,1}^2=m)=e^{-2.4}{(1.1)}^f{(1.3)}^m{(f!m!)}^{-1}$$

(10)

Thus, we are considering two bivariate Poisson laws as offspring distributions. In fact, Poisson probability distribution is very used to describe the probabilistic evolution of biological species. From (9) and (10) we deduce,

$${\mu }^1=(1.3,1.2),{\Delta }^1=\left(\begin{array}{cc}1.3& 0\\ 0& 1.2\end{array}\right)$$

$${\mu }^2=(1.1,1.3),{\Delta }^2=\left(\begin{array}{cc}1.1& 0\\ 0& 1.3\end{array}\right)$$

Offspring distribution $P_1$ favors the birth of females, with a ratio females/males of the means equal to $1.083$. This ratio has a value of $0.847$ for the offspring distribution $P_2$ that consequently favors the birth of males.

3. 

In each generation, we assume the following functions ${\phi }_m$ and ${\phi }_r$:

$${\phi }_m(f,m)=1I_{\{f{m}^{-1}>K_2\}}(f,m)+2I_{\{f{m}^{-1}\le K_2\}}(f,m)$$

$${\phi }_r(f,m)=1I_{\{f\le m\}}(f,m)+2I_{\{f>m\}}(f,m)$$

$I_A$ being the indicator function of the set A and $K_2$ representing a suitable threshold for the ratio females/males.

As illustration, taking, e.g., $X_0=(300,80)$, $K_1=0.75$ and $K_2=1.05$, applying the computing programs we have implemented through the statistical software R, ([15]), we have simulated data for a total number of 30 generations, see

Table 1. Females and males ($X_n$), mating strategy ($l_n$), couples ($Z_n=L_{{\phi }_m\left(X_n\right)}\left(X_n\right)$) and reproductive strategy ($h_n$) in the successive generations.

Generation	${\mathit{X}}_{\mathit{n}}$	${\mathit{l}}_{\mathit{n}}$	${\mathit{Z}}_{\mathit{n}}$	${\mathit{h}}_{\mathit{n}}$	Generation	${\mathit{X}}_{\mathit{n}}$	${\mathit{l}}_{\mathit{n}}$	${\mathit{Z}}_{\mathit{n}}$	${\mathit{h}}_{\mathit{n}}$
0	(300, 80)	1	225	2	16	(154, 124)	1	115	2
1	(236, 302)	2	226	1	17	(130, 134)	2	100	1
2	(308, 275)	1	231	2	18	(124, 105)	1	93	2
3	(238, 298)	2	223	1	19	(104, 135)	2	101	1
4	(307, 269)	1	230	2	20	(128, 117)	1	96	2
5	(229, 301)	2	225	1	21	(122, 106)	1	91	2
6	(288, 271)	1	216	2	22	(86, 131)	2	98	1
7	(244, 240)	2	180	2	23	(125, 106)	1	93	2
8	(182, 239)	2	179	1	24	(106, 130)	2	97	1
9	(223, 218)	2	163	2	25	(122, 148)	2	111	1
10	(191, 214)	2	160	1	26	(131, 125)	2	93	2
11	(197, 205)	2	153	1	27	(89, 115)	2	86	1
12	(190, 169)	1	142	2	28	(119, 111)	1	89	2
13	(148, 195)	2	146	1	29	(96, 116)	2	87	1
14	(172, 183)	2	137	1	30	(112, 118)	2	88	1
15	(162, 175)	2	131	1

.

From Table 1, we have that:

• $T_1=\{1,3,5,8,10,11,13,14,15,17,19,22,24,25,27,29,30\}$,
• $T_2=\{0,2,4,6,7,9,12,16,18,20,21,23,26,28\}$

Hence, by (6), we determine the following estimates for ${\mu }^h,h=1,2$,

$$\widehat{{\mu }^1}=(1.266,1.203),\widehat{{\mu }^2}=(1.070,1.291)$$

(11)

From (11),

$$\underset{h=1,2}{max}\left\{\underset{i=1,2}{max}\left\{\right|\widehat{{\mu }_i^h}-{\mu }_i^h\left|\right\}\right\}=0.034$$

This value indicates good accuracy for the obtained estimates. See also Figure 1.

3. Probability Model with Immigration

In this section, the previous two-sex probability model is extended including immigration of females and males from external populations. On the probability space $(\Omega ,\mathcal{A},P)$ we now introduce the sequence ${\left\{Y_n\right\}}_{n=0}^{\infty }$, $Y_n=(F_n,M_n)$ representing the number of female and male individuals in the population at generation n. Initially we assume $Y_0=(F_0,M_0)\in {\mathbb{N}}_{+}^2$. As in model (1), given that $Y_n=y\in {\mathbb{N}}^2$, then $L_{l_n}$ and $P_{h_n}$ with $l_n:={\phi }_m\left(y\right)$ and $h_n:={\phi }_r\left(y\right)$ are the mating and reproductive strategies at the n-th generation, respectively. At generation $n+1$,

$$Y_{n+1}=(F_{n+1},M_{n+1}):=\sum _{i=1}^{L_{{\phi }_m(y)}(y)}(F_{n,i}^{{\phi }_r(y)},M_{n,i}^{{\phi }_r(y)})+(F_{n+1}^I,M_{n+1}^I),n\in \mathbb{N}$$

(12)

where $F_{n+1}^I$ ($M_{n+1}^I$) represents the number of immigrant females (males). It is assumed that, ${\left\{(F_n^I,M_n^I)\right\}}_{n=1}^{\infty }$ is a sequence of i.i.d non-negative variables (defined on $(\Omega ,\mathcal{A},P)$) independent of $(F_{0,1}^h,M_{0,1}^h),h\in {\mathbb{N}}_r$. The probability distribution (immigration distribution) of $(F_1^I,M_1^I)$ will be denoted by ${\left\{q_{(f,m)}\right\}}_{(f,m)\in \mathbb{N}}$, $q_{(f,m)}:=P(F_1^I=f,M_1^I=m)$.

From (12), given $y_0,\dots ,y_n,y_{n+1}\in {\mathbb{N}}^2$, using that ${\left\{(F_n^I,M_n^I)\right\}}_{n=1}^{\infty }$ is a sequence of i.i.d random vectors independent of $(F_{0,1}^h,M_{0,1}^h)$, $h\in {\mathbb{N}}_r$,

$$\begin{array}{cc}\hfill P(Y_{n+1}=y_{n+1}\mid Y_0=y_0,\dots ,Y_n=y_n)& =P\left(\sum _{i=1}^{L_{{\phi }_m\left(y_n\right)}\left(y_n\right)}(F_{n,i}^{{\phi }_r\left(y_n\right)},M_{n,i}^{{\phi }_r\left(y_n\right)})+(F_{n+1}^I,M_{n+1}^I)=y_{n+1}\right)\hfill \\ \hfill & =P(Y_{n+1}=y_{n+1}\mid Y_n=y_n)\hfill \end{array}$$

Again, the transition probabilities are independent of the generation n considered. In fact, for $y,z\in {\mathbb{N}}^2$,

$$P(Y_{n+1}=z\mid Y_n=y)=P\left(\sum _{i=1}^{L_{{\phi }_m\left(y\right)}\left(y\right)}(F_{n,i}^{{\phi }_r\left(y\right)},M_{n,i}^{{\phi }_r\left(y\right)})+(F_{n+1}^I,M_{n+1}^I)=z\right)=\sum _{z_1,\cdots ,z_d,w\in {\Delta }_z^{*}}\prod _{i=1}^d{p}_{z_i}^{{\phi }_r\left(y\right)}{q}_w$$

where $d=L_{{\phi }_m\left(y\right)}\left(y\right)$ and ${\Delta }_z^{*}:=\{z_1,\dots ,z_d,w\in {\mathbb{N}}^2:{\displaystyle \sum _{i=1}^d}{z}_i+w=z\}$. Thus, ${\left\{Y_n\right\}}_{n=0}^{\infty }$ is a homogeneous Markov chain with state space ${\mathbb{N}}^2$.

In what follows, we provide analogous results to Propositions 1–3 for this new class of two-sex branching processes with immigration of females and males.

For $s,t\in [0,1]$, let $\varphi (s,t):=E\left[s^{F_1^I}{t}^{M_1^I}\right]$, $h_n(s,t):=E\left[s^{F_n}{t}^{M_n}\right],n\in \mathbb{N}$, be the p.g.f. of $(F_1^I,M_1^I)$ and $Y_n=(F_n,M_n)$, respectively. We have that $h_0(s,t)=s^{F_0}{t}^{M_0}$. The general expression for $h_n,n\in {\mathbb{N}}_{+}$, is given in the following result.

Proposition 4.

For $n\in \mathbb{N}$,

$$h_{n+1}(s,t)=\varphi (s,t)E\left[{\left(f_{{\phi }_r\left(Y_n\right)}(s,t)\right)}^{L_{{\phi }_m\left(Y_n\right)}\left(Y_n\right)}\right],s,t\in [0,1]$$

Proof. 

Given $n\in \mathbb{N}$,

$$\begin{array}{cc}\hfill h_{n+1}(s,t)& =E\left[s^{F_{n+1}}{t}^{M_{n+1}}\right]=E\left[E[s^{F_{n+1}}{t}^{M_{n+1}}\mid Y_n]\right]\hfill \\ \hfill & =\sum _{y\in {\mathbb{N}}^2}E\left[s^{\sum _{i=1}^{L_{{\phi }_m\left(y\right)}\left(y\right)}{F}_{n,i}^{{\phi }_r\left(y\right)}+F_{n+1}^I}{t}^{\sum _{i=1}^{L_{{\phi }_m\left(y\right)}\left(y\right)}{M}_{n,i}^{{\phi }_r\left(y\right)}+M_{n+1}^I}\right]P(Y_n=y)\hfill \\ \hfill & =\sum _{y\in {\mathbb{N}}^2}E\left[s^{F_1^I}{t}^{M_1^I}\right]{\left(E\left[s^{F_{0,1}^{{\phi }_r\left(y\right)}}{t}^{M_{0,1}^{{\phi }_r\left(y\right)}}\right]\right)}^{L_{{\phi }_m\left(y\right)}\left(y\right)}P(X_n=y)\hfill \\ \hfill & =\varphi (s,t)E\left[{\left(f_{{\phi }_r\left(Y_n\right)}(s,t)\right)}^{L_{{\phi }_m\left(Y_n\right)}\left(Y_n\right)}\right],s,t\in [0,1].\hfill \end{array}$$

□

For the next result, we assume again that $L_l$, $l\in {\mathbb{N}}_m$, are superadditive functions.

Proposition 5.

Assume $y_0,z_0\in {\mathbb{N}}_{+}^2$ such that $p_{y_0}^h>0$, $q_{z_0}>0$, $L_l\left(y_0\right)>1$, $h\in {\mathbb{N}}_r$, $l\in {\mathbb{N}}_m$. Given $y\in {\mathbb{N}}^2$:

(a) 

There exists $y^{\prime }\in {\mathbb{N}}^2$, with $L_{{\phi }_m\left(y^{\prime }\right)}\left(y^{\prime }\right)>L_{{\phi }_m\left(y\right)}\left(y\right)$, verifying that $y^{\prime }\in C_{y_0}$.

(b) 

$z_0\in C_y$.

Proof. 

(a)

Let us consider the sequence ${\left\{y_n\right\}}_{n=0}^{\infty }$, where

$$y_{n+1}:=y_0{L}_{{\phi }_m\left(y_n\right)}\left(y_n\right)+z_0,n\in \mathbb{N}$$

By the superadditivity of $L_l,l\in {\mathbb{N}}_m$,

$$\begin{array}{cc}\hfill L_{{\phi }_m\left(y_{n+1}\right)}\left(y_{n+1}\right)& =L_{{\phi }_m\left(y_{n+1}\right)}(y_0{L}_{{\phi }_m\left(y_n\right)}\left(y_n\right)+z_0)\hfill \\ \hfill & =L_{{\phi }_m\left(y_{n+1}\right)}\left(\sum _{i=1}^{L_{{\phi }_m\left(y_n\right)}\left(y_n\right)}{y}_0+z_0\right)\hfill \\ \hfill & \ge \sum _{i=1}^{L_{{\phi }_m\left(y_n\right)}\left(y_n\right)}L{\phi }_m\left(y_{n+1}\right)\left(y_0\right)+L_{{\phi }_m\left(y_{n+1}\right)}\left(z_0\right)\hfill \\ \hfill & =L_{{\phi }_m\left(y_n\right)}\left(y_n\right)L{\phi }_m\left(y_{n+1}\right)\left(y_0\right)+L_{{\phi }_m\left(y_{n+1}\right)}\left(z_0\right)\hfill \\ \hfill & >L_{{\phi }_m\left(y_n\right)\left(y_n\right)}\hfill \end{array}$$

Hence ${\left\{L_{{\phi }_m\left(y_n\right)\left(y_n\right)}\right\}}_{n=0}^{\infty }\nearrow \infty$. Thus, given $y\in {\mathbb{N}}^2$, there exists $y_n$ such that $l_{{\phi }_m\left(y_n\right)}\left(y_n\right)>L_{{\phi }_m\left(y\right)}\left(y\right)$. If for some $l\in {\mathbb{N}}_{+}$, $Y_l=y_0$, then:

$$P(Y_{l+n}=y_n\mid Y_l=y_0)\ge \prod _{i=0}^{n-1}P(Y_{l+i+1}=y_{i+1}\mid Y_{l+i}=y_i)=\prod _{i=0}^{n-1}P(Y_{l+1}=y_{i+1}\mid Y_l=y_i)$$

(13)

Now, using that $y_{i+1}=y_0{L}_{{\phi }_m\left(y_i\right)}\left(y_i\right)+z_0$,

$$\begin{array}{cc}\hfill P(Y_{l+1}=y_{i+1}\mid Y_l=y_i)& =P\left(\sum _{j=1}^{L_{{\phi }_m\left(y_i\right)}\left(y_i\right)}(F_{l,j}^{{\phi }_r\left(y_i\right)},M_{l,j}^{{\phi }_r\left(y_i\right)}\right)+(F_{l+1}^I,M_{l+1}^I)=y_{i+1})\hfill \\ \hfill & \ge {\left(p_{y_0}^{{\phi }_r\left(y_i\right)}\right)}^{L_{{\phi }_m\left(y_i\right)}\left(y_i\right)}{q}_{z_0}>0\hfill \end{array}$$

Therefore, from (13),

$$P(Y_{l+n}=y_n\mid Y_l=y_0)\ge {\left(p_{y_0}^{{\phi }_r\left(y_i\right)}\right)}^{{\displaystyle \sum _{i=0}^{n-1}}{L}_{{\phi }_m\left(y_i\right)}\left(y_i\right)}{\left(q_{z_0}\right)}^{n-1}>0$$

We deduce that $y_n\in C_{y_0}$ and the result is proved.

(b)

If for some $n\in \mathbb{N}$, $Y_n=y$, then

$$\begin{array}{cc}\hfill P(Y_{n+1}=z_0\mid Y_n=y)& =P\left(\sum _{j=1}^{L_{{\phi }_m\left(y\right)}\left(y\right)}(F_{n,j}^{{\phi }_r\left(y\right)},M_{n,j}^{{\phi }_r\left(y\right)})+(F_{n+1}^I,M_{n+1}^I)=z_0\right)\hfill \\ \hfill & \ge {\left(p_{(0,0)}^{{\phi }_r\left(y\right)}\right)}^{L_{{\phi }_m\left(y\right)}\left(y\right)}{q}_{z_0}>0\hfill \end{array}$$

□

Let us denote by ${\mu }^I$ and ${\Delta }^I$ the mean vector and the covariance matrix of $(F_1^I,M_1^I)$, respectively. Furthermore, for $n\in \mathbb{N}$ and $y\in {\mathbb{N}}^2$, let ${\eta }_{n+1}^y$ and ${\Gamma }_{n+1}^y$ the mean vector and covariance matrix, respectively, of $Y_{n+1}$ given that $Y_n=y$. From Proposition 4, it can be checked that, independently of n:

$$E[s^{F_{n+1}}{t}^{M_{n+1}}\mid Y_n=y]=\varphi (s,t){\left(f_{{\phi }_r\left(y\right)}(s,t)\right)}^{L_{{\phi }_m\left(y\right)}\left(y\right)},s,t\in [0,1]$$

$${\eta }_{n+1}^y=L_{{\phi }_m\left(y\right)}\left(y\right){\mu }^{{\phi }_r\left(y\right)}+{\mu }^I,{\Gamma }_{n+1}^y=L_{{\phi }_m\left(y\right)}\left(y\right){\Delta }^{{\phi }_r\left(y\right)}+{\Delta }^I$$

(14)

We now consider the estimation of ${\mu }^h,h\in {\mathbb{N}}_r$ and ${\mu }^I$. We will assume that, for some $n\in {\mathbb{N}}_{+}$, we know the observations of the variables:

$$Y_0,L_{{\phi }_m\left(Y_k\right)}\left(Y_k\right),Y_{k+1},Y_{k+1}^{I}k=0,\dots ,n$$

(15)

where, by simplicity $Y_{k+1}^I:=(F_{k+1}^I,M_{k+1}^I)$. Clearly, we can propose as estimator for ${\mu }^I$, based on the data sample (15),

$$\widehat{{\mu }^I}={(n+1)}^{-1}\sum _{k=0}^n{Y}_{k+1}^I$$

(16)

For each $h\in {\mathbb{N}}_r$, let $T_h^{*}:=\{k\in \{0,\dots ,n\}:{\phi }_r\left(Y_k\right)=h\}$ be the set of generations (until the generation n) where $P_h$ has been the underlying reproductive strategy.

Proposition 6.

Given $h\in {\mathbb{N}}_r$ such that $T_h^{*}\ne \varnothing$ and $\sum _{k\in T_h^{*}}{L}_{{\phi }_m\left(Y_k\right)}\left(Y_k\right)>0$, a conditional moment-based estimator for ${\mu }^h$ using the data sample (15), is given by:

$$\widehat{{\mu }^h}={\left(\sum _{k\in T_h^{*}}{L}_{{\phi }_m\left(Y_k\right)}\left(Y_k\right)\right)}^{-1}\sum _{k\in T_h^{*}}(Y_{k+1}-Y_{k+1}^I)$$

(17)

Proof. 

From (14),

$$E[Y_{k+1}\mid Y_k]=L_{{\phi }_m\left(Y_k\right)}\left(Y_k\right){\mu }^{{\phi }_r\left(Y_k\right)}+{\mu }^{I}a.s.$$

Hence, by moment estimation procedure, we propose as estimate for ${\mu }^{{\phi }_r\left(Y_k\right)}$, based on the observations of $L_{{\phi }_m\left(Y_k\right)}\left(Y_k\right)$ (assumed positive), $Y_{k+1}$ and $Y_{k+1}^I$,

$$\overline{{\mu }_{\left(k\right)}^{{\phi }_r\left(Y_k\right)}}={\left(L_{{\phi }_m\left(Y_k\right)}\left(Y_k\right)\right)}^{-1}(Y_{k+1}-Y_{k+1}^I),k=0,\dots ,n$$

(18)

It can be verified that,

$$E\left[\overline{{\mu }_{\left(k\right)}^{{\phi }_r\left(Y_k\right)}}\mid L_{{\phi }_m\left(Y_k\right)}\left(Y_k\right)>0\right]={\mu }^{{\phi }_r\left(Y_k\right)}$$

Taking into account (18), an appropriate estimator for ${\mu }^h$, based on the data sample (15), is given by:

$$\widehat{{\mu }^h}=\sum _{k\in T_h^{*}}{\gamma }_k^h\overline{{\mu }_{\left(k\right)}^h}$$

(19)

where $\sum _{k\in T_h^{*}}{\gamma }_k^h=1$. Taking ${\gamma }_k^h\propto L_{{\phi }_m\left(Y_k\right)}\left(Y_k\right)$, we deduce,

$${\gamma }_k^h=\sum _{k\in T_h^{*}}{\left(L_{{\phi }_m\left(Y_k\right)}\left(Y_k\right)\right)}^{-1}{L}_{{\phi }_m\left(Y_k\right)}\left(Y_k\right)$$

Hence, from (18) and (19), we obtain Expression (17). □

Example 2.

Let the two-sex probability model (1) considered in Example 1. We now assume that, in each generation, immigrant females and males enter the population from other populations according to a certain probability distribution, for example, the trinomial distribution:

$$q_{(f,m)}=(50!){(f!m!(50-f-m)!)}^{-1}{\left(0.4\right)}^f{\left(0.4\right)}^m{\left(0.2\right)}^{50-f-m}$$

$$f,m\in \{0,1,\dots ,50\},f+m\le 50$$

We deduce that,

$${\mu }^I=(20,20),{\Delta }^I=\left(\begin{array}{cc}12& -8\\ -8& 12\end{array}\right)$$

As illustration, taking $Y_0=(300,80)$, $K_1=0.75$ and $K_2=1.05$, we have simulated data for a total number of 30 generations, see

Table 2. Females and males ($Y_n$), mating strategy ($l_n$), couples ($Z_n=L_{{\phi }_m\left(Y_n\right)}\left(Y_n\right)$), immigrant females and males ($Y_n^I$) and reproductive strategy ($h_n$) in the successive generations.

Generation	${\mathit{Y}}_{\mathit{n}}$	${\mathit{l}}_{\mathit{n}}$	${\mathit{Z}}_{\mathit{n}}$	${\mathit{Y}}_{\mathit{n}}^{\mathit{I}}$	${\mathit{h}}_{\mathit{n}}$	Generation	${\mathit{Y}}_{\mathit{n}}$	${\mathit{l}}_{\mathit{n}}$	${\mathit{Z}}_{\mathit{n}}$	${\mathit{Y}}_{\mathit{n}}^{\mathit{I}}$	${\mathit{h}}_{\mathit{n}}$
0	(300, 80)	1	225	(0, 0)	2	16	(379, 356)	1	284	(20, 20)	2
1	(250, 343)	2	257	(17, 23)	1	17	(320, 396)	2	297	(18, 16)	1
2	(389, 314)	1	291	(24, 15)	2	18	(427, 384)	1	320	(16, 17)	2
3	(354, 387)	2	290	(23, 17)	1	19	(358, 428)	2	321	(20, 19)	1
4	(425, 366)	1	318	(17, 25)	2	20	(446, 386)	1	334	(18, 21)	2
5	(369, 411)	2	308	(14, 28)	1	21	(401, 455)	2	341	(24, 17)	1
6	(410, 369)	1	307	(17, 24)	2	22	(459, 452)	2	339	(19, 22)	2
7	(379, 412)	2	309	(22, 19)	1	23	(408, 465)	2	348	(20, 25)	1
8	(453, 411)	1	339	(17, 22)	2	24	(491, 419)	1	368	(19, 21)	2
9	(389, 444)	2	333	(17, 24)	1	25	(432, 501)	2	375	(24, 20)	1
10	(484, 410)	1	363	(21, 23)	2	26	(514, 433)	1	385	(20, 21)	2
11	(453, 445)	2	333	(21, 19)	2	27	(437, 509)	2	381	(18, 23)	1
12	(426, 425)	2	318	(27, 17)	2	28	(520, 524)	2	393	(20, 20)	1
13	(369, 408)	2	306	(16, 22)	1	29	(497, 509)	2	381	(23, 18)	1
14	(389, 374)	2	280	(13, 22)	2	30	(464, 496)	2	372	(22, 16)	1
15	(306, 375)	2	281	(14, 24)	1

.

From Table 2, we have that:

• $T_1^{*}=\{1,3,5,7,9,13,15,17,19,21,23,25,27,28,29,30\}$,
• $T_2^{*}=\{0,2,4,6,8,10,11,12,14,16,18,20,22,24,26\}$

Hence, by (16) and (17),

$$\widehat{{\mu }^I}=(18.633,20.133),\widehat{{\mu }^1}=(1.313,1.198),\widehat{{\mu }^2}=(1.115,1.272)$$

We obtain,

$$\underset{i=1,2}{max}\left\{|\widehat{{\mu }_i^I}-{\mu }_i^I|\right\}=1.367,\underset{h=1,2}{max}\left\{\underset{i=1,2}{max}\left\{\right|\widehat{{\mu }_i^h}-{\mu }_i^h\left|\right\}\right\}=0.028$$

These values indicate good accuracy for the proposed estimates. See also Figure 2 and Figure 3.

4. Conclusions

In this research, we have focused attention to the mathematical modeling of the population dynamics in biological species with sexual reproduction. We have considered the possibility of multiple mating and reproductive strategies, thereby continuing the research line initiated in previous papers. Several probabilistic and statistical contributions have been derived. In particular, general expressions for the probability generating functions associated with the variables of interest in the underlying probability model have been deduced (Proposition 1), some properties about the behavior of the states of the process have been studied (Proposition 2) and estimates for the mean vectors of the offspring distributions have been proposed (Proposition 3). This class of two-sex branching models has been generalized by considering immigration of females and males from external populations. The previous results have been then extended to this new class of models with immigration (Propositions 4–6). As illustration, for both classes of two-sex models, simulated examples have been presented.

Some questions for future research are, e.g., consider alternative inferential procedures in order to estimate the main parameters governing the reproduction phase; determine the probability distribution associated with the number of generations elapsed before the possible extinction of the population; or explore potential applications of the investigated two-sex models in phenomena of ecological and environmental interest, for example, in mathematical modeling of the phenomenon concerning populating or re-populating a certain habitat with some semelparous species.