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Researchers should cite this work as follows:

• Sebeysan, G. S., Farago, I. (2023). 2015-Sebesyen-Farago-Invasive Species Model with Linear Rat Harvesting on Easter Island. SIMIODE, QUBES Educational Resources. doi:10.25334/N9SG-AC55

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Sebesyen, G. S. and I Farago. 2015. Invasive Species Model with Linear Rat Harvesting on Easter Island. J Appl Computat Math. 4(6): 1-6.

https://www.researchgate.net/publication/294108452_Invasive_Species_Model_with_Linear_Rat_Harvesting_on_Easter_Island . Accessed29 March 2023.

From the paper we quote:

The history of the civilization on Easter Island has long interested archaeologists. The Easter Island is located in the Pacific Ocean, at the southeastern point of the Polynesian Triangle. It is famous for the culture and monumental stone statues, so called Moai. We know from archeological records, that at the time of the initial settlement, the island had many species of trees, e.g., palm species which grew up to 15 meters or more. In 1786 comte de La Perouse’s visited to the island and found only 2000 inhabitants and no trees. People used the trees for construction and transportation of statues. An other factor was the extinction of plants species, was the appearance of the Polynesian rat. Studies have shown the dramatically effect of the rats in the ecosystem. These factors caused the population to collapse.

In 2008, Basener, Brooks, Radin and Wiandt presented an article in that, they created a mathematical model [1]. The invasive species model is a system of three differential equations, which describes the relations between the people, trees and rats. In this model, it is assumed that the amount of the resources available for the people is proportional to the number of trees. The growth rate of the human population is defined by the logistic equation. Analogically, the growth rate of the rat population is defined by the similar logistic equation where the carrying capacity is the amount of the trees. In the equation for the rat population we assume that the rats eat the seeds of the trees and the humans also decrease the amount of the trees. The stability property of this model is investigated both theoretically and numerically.

In this paper we suggest a natural modification of this model. Namely, we will investigate the case where the amount of the rats is decreased due to some external factor, e.g., exterminations by the people. We investigated the effect of this new added factor to the stability property of the model. We show that the system has a conditionally stable equilibrium point, and in this case the three populations live together. We also made numerical simulations with explicit numerical solvers which supports the theoretical result, namely, that the amount of the rats decreases because of the harvesting. The paper is organized as follows.

We gave the description of the invasive species model, given by Basener, Brooks, Radin and Wiandt. We define the equilibrium points for the system and investigate their stability. We analyze their stability property, by using Roughgarden theorem. We construct the discrete models by using the explicit Euler method on uniform mesh. We examine four different cases and the numerical results confirm our theoretical results. The paper is finished with some conclusion. We added an Appendix to the paper which includes some technically complex calculations in order to check the conditions of the stability.

Keywords: differential equation, model, applications, harvest, invasive, invasive species, rats, Easter Island, stability, carrying capacity