# 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
The number of adults (males and females) is affected by factors such as the success of metamorphosis, the proportion of sex differentiation, and the mortality rate of adults. Sex differentiation can be expressed as the proportion of females \(P_{f}\) and the proportion of males \(P_{m}\) in the adult population, while we have  
\[P_{m}+P_{f}=1 \tag{2}\]  
From this, we used a set of differential equations with time as the independent variable to describe the number of lampreys at each stage. We adopted Assumption 3 (overlook the impact of disease spread and movement within and outside populations on the size of the population) and Assumption 5 (individuals at an identical developmental phase in a population are considered equal here). \[\frac{dN_{1}}{dt}=\gamma\delta_{3}N^{*}-\delta_{1}N_{1}-\mu_{1}N_{1} \tag{3}\]  
\[\frac{dN_{2}}{dt}=\delta_{1}N_{1}-\delta_{2}N_{2}-\mu_{2}N_{2} \tag{4}\]  
\[\frac{dN_{3}}{dt}=P_{m}\delta_{2}N_{2}-\delta_{3}N^{*}-\mu_{3}N_{3} \tag{5}\]  
\[\frac{dN_{4}}{dt}=P_{f}\delta_{2}N_{2}-\delta_{3}N^{*}-\mu_{4}N_{4} \tag{6}\]  
\[N^{*}=min(N_{3},N_{4}) \tag{7}\]  
where \(\gamma\) is the hatching success rate, \(\delta_{1},\ \delta_{2},\ \delta_{3}\) are the maturation rate from fertilized eggs to juveniles, the maturation rate from juveniles to adults, and the mating rate of adults, and \(\mu_{1},\ \mu_{2},\ \mu_{3},\ \mu_{4}\) are the mortality rates of lamprey eggs, juveniles, and adult males and females, respectively. Here, \(N^{*}\) is the lower number of males and females, which determines the likelihood of mating. This setting ensures that the number of mating events is limited by the sex ratio balance. Relationship between sex ratio and food availability under sex-determined mechanisms  
Lampreys' sex determination mechanism is environmentally delicate and, in contrast to numerous other species, does not rely exclusively on genetic elements. Specifically, the sex of the lampreys is not fixed at fertilization but is influenced by specific conditions in the environment, especially food availability. This mechanism of sex determination is characterized by the following:  
Association between growth rate and sex differentiation: During the larval stage of lampreys, the growth rate of an individual affects its eventual sex. This means that sex determination is directly related to the environmental conditions encountered by the individual during the initial phase of development. Role of food availability: Food availability is considered a key factor in determining the growth rate of larvae. Larvae in environments abundant in food exhibit quicker growth, typically leading to a greater female ratio. Conversely, in environments with limited food availability, larvae exhibit slower growth and a comparatively larger male population. Variability in sex ratio: Since sex is determined based on environmental conditions, this leads to significant variability in the sex ratio of lampreys under different environmental conditions. For example, when food availability is low, the proportion of males can reach about 78% of the population, whereas in environments abundant in food, the proportion of males may drop to about 56% [1]. Therefore, we first constructed a sex ratio function for lampreys. Figure 3:0  
Figure 3:1  
Equations (8) and (9) describe the relationship between sex ratio and food availability per unit of larva. \(P_{m}\) is the proportion of males in the total adult population. \(x\) denotes the average amount of food available per larva, where \(f\) is the total amount of food available and \(N_{2}\) is the number of lampreys in the larval stage. We adopted Assumption 1 (food supply is the main factor influencing the growth rate of sea eel larvae). \(F(x)\) is a decreasing function reflecting the ecological trend that greater food availability leads to a lower male proportion and a higher female proportion. The specific form of the sex ratio function is as follows:  
Figure 3:2  
The effect of food quantity on the sex ratio is regulated by parameters \(a_{1}\) and \(b_{1}\).