# 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 function may reflect the tendency for the proportion of males to decrease and the proportion of females to increase as food availability increases. Since food availability is dynamic, we developed an equation for food availability over time  
Figure 3:3  
The initial gender composition of the adaptive mechanism population is  
\[\left\{ \begin{array}{c}P_{m}=F\left(\frac{f_{0}}{N_{2}(0)}\right)\\ P_{f}=1-P_{m}\\ N_{2,male}(0)=P_{m}\cdot N_{2}(0)\\ N_{2,female}(0)=P_{f}\cdot N_{2}(0)\\ N_{2,female}(0)+N_{2,male}(0)=N_{2}(0)\end{array} \right. \tag{12}\]  
where \(f_{0}\) is the total amount of food available at the initial time, \(N_{2,female}(0)\) is the number of female lampreys in the larval stage at the initial time, \(N_{2,male}(0)\) is the number of male lampreys in the larval stage at the initial time, and \(N_{2}(0)\) is the the number of lampreys in the larval stage at the initial time. Dominant population model based on competitive mechanisms  
The basic mathematical model of the competition between two species is constructed below. Let two kinds of groups \(i,\ j\), in the same environment rely on the same limited resources to survive. Set the moment \(t\) when the two species' total numbers were \(N_{i}(t)\) and \(N_{j}(t)\), the growth of the population is subject to their laws, the natural growth rate of \(\rm r_{i}\) and \(r_{j}\), respectively, when the other side of the extinction of the number of survival is, respectively, \(K_{i}\) and \(K_{j}\). In this context, \(N_{i,larau}(0)\) is the \(N_{2}(0)\) referenced in Equation (12). \[N_{i}(0)=N_{i,agg}(0)+N_{i,larau}(0)+N_{i,adult}(0) \tag{13}\]  
Set the initial moment when both populations are small. If the consumption of resources by the individuals of the second population is \(a_{i}\) times the consumption of the individuals of the first population, then the growth rate of the first species group is  
\[r_{i}\cdot\left(1-\frac{N_{i}+a_{i}N_{j}}{K_{j}}\right)N_{i} \tag{14}\]  
Similarly, if individuals of the first population consume \(a_{j}\) times more resources than individuals of the second population, the growth rate of the second species group is  
\[r_{j}\cdot\left(1-\frac{N_{j}+a_{j}N_{i}}{K_{j}}\right)N_{j} \tag{15}\]  
The growth rate of each population (\(r_{i}\) and \(r_{j}\)) reflects the natural ability to grow in the absence of competitive pressures. This rate depends on the reproduction rate, mortality rate, and viability of individuals. Here, we have employed basic Assumption 4 (limited environmental carrying capacity). \(K_{i}\) and \(K_{j}\) denote the maximum population size that each population can reach in the absence of competition, i.e., the carrying capacity of the environment. \(a_{i}\) and \(a_{j}\) denote the resource consumption between two populations. This ratio determines the amount of competitive pressure one population exerts on the other. Thus, the total number of two populations \(N_{i}(t)\) and \(N_{j}(t)\) satisfy the following set of differential equations  
\[\left\{ \begin{aligned} &\frac{dN_{i}}{dt}=r_{i}\cdot\left(1-\frac{N_{i}+a _{i}N_{j}}{K_{j}}\right)N_{i}\\ &\frac{dN_{j}}{dt}=r_{j}\cdot\left(1-\frac{N_{j}+a_{j}N_{j}}{K_{j }}\right)N_{j}\end{aligned} \right. \tag{16}\]  
Introducing  
\[b_{ii}=\frac{r_{i}}{K_{i}},b_{ij}=\frac{r_{i}a_{i}}{K_{i}},b_{ji}=\frac{r_{i}a _{j}}{K_{j}},b_{jj}=\frac{r_{j}}{K_{j}} \tag{17}\]  
where \(b_{ii}\), \(b_{ij}\), \(b_{ji}\), and \(b_{jj}\) are competition coefficients that characterize the strength of interactions between populations. These coefficients reflect the degree of influence of