Population Model Slide1 What I want to explain to you about this curve is think about how different populations that share the same ecosystem can interact with each other and actually provide a feedback loop on each other. And there's many cases of this, but the most cited general example is the case when one population wants to eat another population. And so, you have the predator population that likes to eat the prey. So you have the predator and prey interactions. So let's just think about how these populations could interact. the horizontal axis is time. The vertical axis is population. And so, let's just, in our starting point, let's say that our prey is starting out at a relatively high point. Let's say we're right there in time, and let's say for whatever reason, our predator population is relatively low. So, what do we think I going to happen here? Well, at this point, with a low density of predators, it's going to be much easier for them for find a meal, and it's going to be much easier for the prey to get caught. So since it's more easy, it's easier for the predators to find a meal, you can imagine their population starting to increase. But what's going to happen is their population is increasing. Well, it's going to be more likely that they're going to, they prey is going to get caught. There's going to be more of their hunters around, more of their predators around. So that population In going to start decreasing all the way to a point where if the population of the prey gets low enough, the predators are going to have, they're going to start having trouble finding food again, and so that their population might start to decrease, and as their population decreases, what's going to happen to the prey? Well, then, there's going to be less predators around, so they might be able to, their population might start to increase. And so I think you see what's happening. The predator and prey, they can kind of form this cyclic interaction with each other. And what I've just drawn, this is often known as the predator-prey cycle. And I just reasoned through that you can imagine a world where you can have the cycle between predator and prey populations. But you can also run computer simulations that will show this, and even observational data out in the field also shows this. One of the often-cited examples The dynamics of predator-prey interactions using the Lotka-Volterra model. It begins with a scenario where the prey population is high, and the predator population is low. As the predator population increases due to easy access to prey, the prey population starts declining. Eventually, the reduced prey population makes it difficult for predators to find food, causing their population to decrease. With fewer predators, the prey population can then increase again, creating a cyclic interaction known as the predator-prey cycle. This narrative highlights the reciprocal relationship between predator and prey populations, a phenomenon supported by both theoretical reasoning and observations in the field. And this fully applies to the following model, Lynx, and snowshoe hare, so the prey is the snowshoe hare, and the predator is the lynx. Slide2 The Lotka-Volterra equations, or predator-prey equations, consist of a pair of first-order nonlinear differential equations. These mathematical expressions model the dynamics between two interacting species in an ecosystem, typically a predator and its prey. The equations capture the changes in the populations of both species over time, highlighting the cyclic nature of their relationship. The model is widely used in ecological studies to understand the dynamics of predator-prey interactions and the delicate balance within ecosystems. Slide3 Assumption1: The prey population has sufficient food resources, ensuring their adequate supply. Assumption2: The size of the prey population is directly correlated with the quantity of available food, indicating a clear relationship. Assumption3: The population's size is directly linked to the rate of change within that population. Assumption4: Predators exhibit continuous feeding behavior without interruption.