In this video I go over another example on predator-prey systems for population growth of two species, but this time modify the Lotka-Volterra equations to instead model two species that compete or cooperate in obtaining resources. Two different sets of differential equations are presented, and in this example I show the steps involved in determining what kind of model each set is describing. This is done by seeing how the population growth of one species is affected by the increases or decrease of the population of the other species. If it's a mutual increase in population growth whenever either population increases, then this represents a cooperation model, such as a bees or insects pollinating flowers. On the other hand, if the populations of either species decreases with the increase of the other species, then this represents a model that shows two species fighting for the same species, such as a pair of plant-eating animals. This is a very good video to show how to read and understand systems of differential equations and how they can be used to model various real world phenomenon.
Watch on DTube: https://steemit.com/dtube/@mes/rf3bnp67
Watch on BitChute:
Download PDF Notes: https://1drv.ms/b/s!As32ynv0LoaIht4Lx_Db7-g03Axxpg
View Video Notes Below!
Download These Notes: Link is in Video Description.
View These Notes as an Article: https://steemit.com/@mes
Subscribe via Email: http://mes.fm/subscribe
Donate! :) https://mes.fm/donateReuse of My Videos:
- Feel free to make use of / re-upload / monetize my videos as long as you provide a link to the original video.
Fight Back Against Censorship:
- Bookmark sites/channels/accounts and check periodically
- Remember to always archive website pages in case they get deleted/changed.
Join my private Discord Chat Room: https://mes.fm/chatroom
Check out my Reddit and Voat Math Forums:
Buy "Where Did The Towers Go?" by Dr. Judy Wood: https://mes.fm/judywoodbook
Watch my #AntiGravity Video Series: https://mes.fm/antigravity-playlist
Follow My #FreeEnergy Video Series: https://mes.fm/freeenergy-playlist
NOTE: If you don't have time to watch this whole video:
- Skip to the end for Summary and Conclusions
- Play this video at a faster speed.
--TOP SECRET LIFE HACK: Your brain gets used to faster speed. (#Try2xSpeed)
--Try 4X+ Speed by Browser Extensions, HookTube.com, Modifying Source Code.
--Browser Extension Recommendation: https://mes.fm/videospeed-extension- Download and Read Notes.
- Read notes on Steemit #GetOnSteem
- Watch the video in parts.
Example:
Each of the system of differential equations below is a model for two species that either compete for the same resources or cooperate for mutual benefit (flowering plants and insect pollinators, for instance).
Decide whether each system describes competition or cooperation and explain why it is a reasonable model.
- Ask yourself what effect an increase in one species has on the growth rate of the other.
Solution
a) Let's take a look at the first differential equation:
The growth rate of x consists of a logistic model term and a xy term, representing encounters with y.
The logistic model term represents the growth of x independently from y, and up to a carrying capacity of 200.
Since xy term is added to the logistic term, this means that as x and/or y increase, the overall growth rate of x also increases.
Now, let look at the second differential equation:
Since every term is positive, the growth rate of y increases whenever x and/or y increases.
Thus this system of differential equations is describing a cooperation model.
b) Let's take a look at the first differential equation:
Similarly to part a) the logistic term shows that in the absence of y, that x increases to a carrying capacity of 750.
Since we are subtracting the xy term, this indicates that the growth rate of x decreases as the number of encounters of x and y increases.
This behavior is the exact same for the second differential equation as well:
The growth rate of y grows independently of x until a carrying capacity of 2500.
But as the number of encounters of x and y increase, represented by the xy term, then the growth rate decreases.
Thus, this system of differential equations is describing a competition model in which both populations are competing for the same resource.