1 / 8

Section 6.3 – The Logistic Equation

Section 6.3 – The Logistic Equation. Population Growth. We have seen a model for the growth of a population ( P ) when This model represents exponential growth:. Now use separation of variables to find the general solution.

garima
Download Presentation

Section 6.3 – The Logistic Equation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Section 6.3 – The Logistic Equation

  2. Population Growth We have seen a model for the growth of a population (P) when This model represents exponential growth: Now use separation of variables to find the general solution. Since C is arbitrary, ±eCrepresents an arbitrary nonzero number. We can replace it with C:

  3. Population Growth/Decay Populations may grow exponentially over short periods of time, but it should be clear that no population can increase without limit. (Imagine a bunny population or bacteria in a petri dish.) Population biologists use a variety of other differential equations that take into account limitations to growth. One widely used is based on the logistic differential equation: where k>0, P>0, and L>0 Let’s investigate this differential equation.

  4. Analytical Investigation of the Logistic Model Consider how the following limits affect the differential equation: The population is very small: Exponential Growth The population is close to the constant L: Nearly no Growth The population is very large: The Growth Rate is Decreasing

  5. Graphical Investigation of the Logistic Model Consider the slope field for : When is the population increasing? NOTE: Even though more are dying, they are still reproducing. P(t) When the population (P) is less than L. When is the population decreasing? When the population (P) is greater than L. L When is the population constant? When the population (P) is equal to L. What happens as time increases? The population (P) always gets closer to L. What does L represent graphically? L is a vertical asymptote. What does L represent contextually? L is the carrying capacity. (The equilibrium solution.) t

  6. A Different Graphical Approach It is helpful to look at a graph of versus : If P is the independent variable and dP/dt is the dependent variable, what is the shape of the graph? L/2 is also an inflection point A concave down Parabola. What are the P intercepts of the graph? What does the population need to be for the growth to be increasing at the fastest rate? Only if the graph goes through this point. At the vertex:

  7. Logistic Model The rate of change of the size of a population P is governed by the logistic model when the population grows in the presence of limited resources.If P represents the number of organisms in a population at time t, the model is represented by the following differential equation: is a positive constant. is the carrying capacity () reaches its maximum at .

  8. Example The population of a species satisfies the logistic differential equation , where the initial population and is the time in years. • What is ? • What population causes to have a maximum value? Does our population ever reach this rate of change? Explain. • When will the population reach 15,000? Explain. • Find the particular solution, , for the initial condition. 10,000 5,000. Yes. The initial population is below 5,000 and it is increasing. No. It can not be greater than the carrying capacity. We do not have the tools to complete this yet.

More Related