# Modeling Population Dynamics and Assessing Ecological Impacts of Lampreys via Sex Ratio Regulation
## 3 Resource-Dependent Sex Ratio Dynamics Model (RDSRD Model)
### Two-Species Competition Mechanism  
This part will focus on creating a model for population competition, integrating the lampreys' life cycle traits and gender determination processes. Population expansion occurs in three distinct phases according to the life cycle  
The life cycle of lampreys is a complex process divided into fourmain stages, each with its own unique physiological and ecological characteristics. These fourstages include the Hatching Stage, Larval Stage, Metamorphosis Stage, Adult Stage. The four-phase life cycle change is shown in Figure 3 below. \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Description** \\ \hline K & Environmental capacity \\ Ni & The \(i\)-th lamprey population \\ \(\delta_{i}\) & Growth rate of the \(i\)-th lamprey population \\ \(\mu_{i}\) & Mortality rate of the \(i\)-th lamprey population \\ \(f\) & Total amount of food \\ \(\eta\) & Environmental Resource Utilization \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notations used in this paper. Figure 3: Growth diagram of the three stages of the lamprey life cycle [11]. The Hatching Stage (HS) is the first stage of the life cycle of lampreys. Lamprey eggs are diminutive yet yield a substantial amount. On each occasion, their egg production ranges from 80,000 to 100,000. Typically, eggs are found clinging to sandy and gravelly surfaces. Soon after emerging, they transform into fry and progress to the larval phase. The Larval Stage (LS) can last up to seven years or more, which is the longest stage of the lampreys' life cycle [12]. During this period, larvae live on the bottom of freshwater streams, usually buried in silt or sand, forming U-shaped or slanted burrows. The Metamorphosis Stage (MS) is relatively short and marks a rapid transition from larva to adult. At the end of this stage, individuals complete sex differentiation, which may be influenced by environmental conditions such as food availability. The Adult Stage (AS) is the final stage of the lampreys' life cycle and is shorter in duration relative to the larval stage. Mature lampreys evolve into predators, consuming tiny fish, invertebrates, and various animals. At maturity, lampreys return to freshwater streams to lay eggs and die shortly thereafter, completing their life cycle [12; 13]. During the modeling process, the metamorphosis stage was ignored because it was in a transitional phase and had a relatively short duration. In conjunction with the stage characterization of sea lampreys, we divided the population growth model into three stages. The relationship between number changes in the three phases is shown in Figure 4 below. After the larva's sex is determined, it transforms into an adult. The adult stage of lampreys is divided into male and female. Consequently, the number of the lamprey population is calculated as  
\[N=N_{1}+N_{2}+N_{3}+N_{4} \tag{1}\]  
where \(N_{1}\), \(N_{2}\), \(N_{3}\), and \(N_{4}\) are the number in the hatching stage, the number in the larval stage, the number of male adults, and the number of female adults, respectively. To build a model reflecting the number of the sea lamprey population, we need to define the pattern of change and the interrelationship of numbers in each stage. The incubation phase (N1) encompasses the transformation of fertilized eggs into larvae. The success rate of hatching can be affected by multiple elements, such as temperature, the intensity  
Figure 4: Relationship between the number changes in the three stages of lampreys. of predation, and the prevailing environmental conditions. Larval stage (N2) refers to the phase post-hatching where larvae undergo growth until their transformation into adults. Number growth during this period can be affected by factors such as food availability, mortality, and growth rate. Adult stage (N3 for males and N4 for females): larvae pass through the metamorphosis stage to become adults, with the sexes differentiating at the end of the metamorphosis stage.