# Modeling Population Dynamics and Assessing Ecological Impacts of Lampreys via Sex Ratio Regulation
## 4 Adaptive Sex Ratio Impact Model (ASRI Model)
### Predation and Prey Model Based on the Lotka-Volterra Equation  
The role played by lampreys in the ecosystem is complex, whereby larvae live at the bottom of freshwater streams and feed mainly on microorganisms and organic matter in filtered water, considered to be parasites that have a significant impact on the ecosystem. Adult lampreys may migrate to different waters, including freshwater and brackish environments, and adult lampreys turn into predators, feeding on small fish, invertebrates, and so on. Concurrently, lampreys serve as a food source in certain global areas, including Scandinavia, the Baltic Sea, and among indigenous communities in North America's Pacific Northwest. Therefore, we examine the role of lampreys as predators and prey in the food chain in the community separately. Figure 9 illustrates the varied functions of lampreys within the ecosystem dynamics.  
Modeling of lampreys as predators  
In this model, we assume that lampreys predominantly prey on a particular fish (the prey). Using the Lotka-Volterra equation [15], we can express this as:  
Population growth equation for the prey(fish):  
\[\frac{dN_{prey}}{dt}=\left(r_{prey}-\alpha_{1}N_{lamp}\right)\cdot N_{prey} \tag{26}\]  
where \(N_{prey}\) is the number of prey, \(r_{prey}\) is its natural growth rate, \(N_{lamp}\) is the number of predators (lampreys), and \(\alpha_{1}\) is the predation rate coefficient. \(r_{prey}N_{prey}dt\) is the naturally increasing number, and \(\alpha_{1}N_{lamp}N_{prey}dt\) is the number of prey that are preyed upon. The quantity of prey being preyed upon is proportional to the number of prey and the predator, because only when predators and prey meet will predation occur.  
Population growth equation for predator (lampreys):  
\[\frac{dN_{lamp}}{dt}=\left(\beta_{1}\alpha_{1}N_{prey}-m_{lamp}\right)\cdot N _{lamp} \tag{27}\]  
where \(\beta_{1}\) is the transformation efficiency coefficient (the efficiency with which a preys transformed into a new individual of the predator) and \(m_{lamp}\) is the natural mortality rate of the predator (lampreys). \(\beta_{1}\alpha_{1}N_{prey}N_{lamp}dt\) is the increase in the number of predators (lampreys) due to the hunting process. The additional coefficient in Equation (27) is because the predators may need to eat several times before they can produce new individuals, thus there is a transformation efficiency coefficient \(\beta_{1}\). \(m_{lamp}N_{lamp}dt\) refers to the number of dead lampreys.  
Modeling of lampreys as prey  
In another scenario, we can consider lampreys as the prey and some larger predatory fish or other aquatic animal as the predator. The model equations can be set up similarly to obtain the following system of equations:  
\[\left\{\begin{array}{l}\frac{dN_{lamp}}{dt}=\Big{(}r_{lamp}-\alpha_{2}N_{ predator}\Big{)}\cdot N_{lamp}\\ \frac{dN_{proatator}}{dt}=\Big{(}\beta_{2}\alpha_{2}N_{lamp}-m_{ predator}\Big{)}\cdot N_{ predator}\end{array}\right. \tag{28}\]  
where \(r_{lamp}\) is the natural growth rate of lampreys, \(\alpha_{2}\) is the predation rate coefficient, and \(N_{predator}\) is the number of predators. \(\beta_{2}\) is the transformation efficiency coefficient, and \(m_{predator}\) is the natural mortality rate of predators.