We study the dynamics of a modified Leslie–Gower one prey–two predators model with competition between predator populations. The model describes complex dynamics in the permanence, global stability and bifurcation. It is shown that there are eight possible equilibrium states. Two equilibrium states, i.e., the extinction of all of the species state and the extinction of both predators state are always unstable, while the other equilibrium states are conditionally locally and globally asymptotically stable. We also analyzed numerically the effect of competition between predators. Our numerical simulations showed that the competition rate of the second-predator may induce the transcritical bifurcation, the saddle-node bifurcation as well as the bi-stability phenomenon. Such numerical results are consistent with the analytical results.

Mathematical reasoning has a clear structure. In mathematical reasoning structure, working memory capacity is considered as something important. This study aimed at knowing the mathematical reasoning structure of junior high school students in solving problems reviewed based on working memory capacity. This research was a qualitative research. This study was carried out in state Junior High School in Makale. The selection of research subjects was done by providing tests to measure students’ working memory capacity. The test used was O-SPAN Test and the researcher chose 1 student with high category working memory capacity and 1 student with low category working memory capacity. Then the researcher gave a test to investigate the reasoning structure. Mathematical reasoning structure of research participant with low working memory capacity showed the stage of problematic situation, strategy choice, conclusion could be done. However, for the stage of strategy is implemented, subject implemented the strategy in one way and could not find a maximum result. Subject with low working memory capacity could provide conclusions but could not give verification arguments showing the strategy used was correct or reasonable and could solve the problem. Mathematical reasoning structure of research subject with high working memory capacity fulfilled the four mathematical reasoning structures namely problematic situation, strategy choice, implemented strategy, and conclusion.

This research was a qualitative research which aimed at describing the understanding of the integral concept of prospective teachers through actions and processes in terms of gender difference. The research subjects were one female student and one male student of Mathematics Education study program. Data were collected through assignment sheets and followed by interviews. After data were analyzed, it was found that the action of the female subject in determining the definite integral was certainly shown by completing the integral task procedurally using the appropriate formula. At the action stage, in determining the total area female subject directly used the integral formula regardless the form and location of the region formed by a curve. The action of male subject in determining the integral was certainly similar to the female subject. It was shown by calculating the integral using the appropriate formula. For the task of determining the total area under the curve, male subject first described the form of the curve and paid attention to the position of the region with respect tox-axis. At the process stage, female subject explained the steps to complete the integral. When explaining how to calculate the total area under the curve, the female subject explained the solution steps regardless the position of the region. Meanwhile, the process stage of male subject was shown by explaining the steps and revealing the reasons for using the formula he used after identifying the form of the given integral. The male subject gave attention to the position of the region with respect tox-axis when calculating the region.

‎In prey-predator models‎, ‎nonlinear interaction between prey and predator populations results in oscillatory behavior that shows the dynamic growth of the populations‎. ‎In the growth process‎, ‎very often both prey and predator share the same resource in their habitat‎. ‎This is an intraguild predation model‎. ‎This study focuses on an intraguild prey-predator model generalized by introducing Holling type III functional response‎. ‎The existence of biologically meaningful equilibria has been investigated‎. ‎The stability analysis of the equilibria has been determined‎. ‎Finally‎, ‎bifurcation and numerical analyses have been presented to illustrate the dynamic behavior of the system‎. ‎Taking the {biotic resource enrichment as the bifurcation parameter}‎, ‎a Hopf bifurcation takes place‎, ‎where solutions with limit cycle behavior appear‎. ‎Varying further the parameter‎, ‎a fold bifurcation of the limit cycle takes place‎, ‎where the unstable limit appeared due to Hopf bifurcation reverses its growing direction and changes its stability‎. ‎Taking the predation rate as the {bifurcation parameter}‎, ‎saddle-node bifurcations take place‎. ‎The existence of stable interior equilibria and stable periodic solutions‎, ‎of which all prey and predator populations and the resource co-exist‎, ‎guarantee the boundedness of the size of the populations and the resource‎. ‎This is good from the conservation of an ecosystem point of view‎.