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--- 2017:groups:g4:start [2017/01/21 05:15] – [Model assumptions] group4
+++ 2017:groups:g4:start [2024/01/09 18:45] (current) – external edit 127.0.0.1
@@ Line 35: / Line 35: @@
   * Along the same lines, can beetles persist preying on ants in the absence of the parasitoid flies?
   * How does the behavior of parasitized ants alter the costs and benefits of predation and parasitism, from the point of view of the ant colony? For instance, if parasitized ants are still productive workers, the cost of being parasitized can be smaller than the benefit provided by predation.
+===== Questions =====
+Do parasitized ants go back to the colony?
+===== References =====
+{{ https://s-media-cache-ak0.pinimg.com/236x/c2/a1/eb/c2a1eb3d2e5a2b5de83d1b2c11528b48.jpg}}
+  * Mathis, K. A., and N. D. Tsutsui 2016 Dead ant walking: a myrmecophilous beetle predator uses parasitoid host location cues to selectively prey on parasitized ants. Proceedings of the Royal Society of London B 283:2016 1281
+  * Video of phorid flies in action: https://www.youtube.com/watch?v=pFVvOo1Qd_8
+====== Results ======
+===== Matrix =====
+<code>
+J =
+[ (r*((m*(b + h))/(k*(a*h - a*g*m*(b + h))) - 1))/((a*g*m*(b + h))/(a*h - a*g*m*(b + h)) + 1) - r*((2*m*(b + h))/(k*(a*h - a*g*m*(b + h))) - 1),       0, -(m*(b + h))/h]
+[                                                                                                -r*((m*(b + h))/(k*(a*h - a*g*m*(b + h))) - 1), - b - h,  (m*(b + h))/h]
+[                                                                                                                                             0,       h,             -m]
+</code>
 ===== Biological scenario and assumptions =====
@@ Line 40: / Line 59: @@
 We investigate two scenarios: one with a specialist beetle feeding exclusively on the parasitised ants and a second scenario with a generalist beetle, which feeds on both types of ants (normal and parasitised).
+===== Análisis de Nicolas =====
+The stability criterion gives a minimum predation rate performed by beetles. If we are over that minimum level, bettles are eating so many infected ants, so that affects flies and normal ants have no problem in converging to the capacity (asymptotic stability).
+If \mu(fly death rate) is greater, then the minimum predation rate is smaller and it is easier to normal ants to converge to capacity. Similar result is true, considering the parameter \gamma (incubation rate).
+If we are below the minimum level of predation rate, then there exists a coexistence equilibrium, so flies have a chance to survive by attacking ants.
+Formally, If \mu (fly death rate),\b (predation rate), are greater then the equilibrium number of ants N^* is bigger. If \alpha (attack rate of flies) is greater, then N^* is smaller.
 ===== Model assumptions =====
@@ Line 64: / Line 91: @@
 ===== Parameter Estimation =====
+Please write down in here where did you get each of the parameters we are using. Thank you!
+Saturday is sensitivity day, hopefully also miracle day.
 ===== Beetles Preference of Parasitized Ants =====
 //What is the criterion and meausure that we shall use here?//
+**Take this with a grain of salt, please.**
+Notice that this model is an extension of the previous one to include the specialization of beetles. We consider the beetles population dynamic and assume it to be very specialist, so that beetles may only grow in the presence of ants. There is a predation term of normal ants by beetles which is mediated by the proportion of normal and parasitized ants. We include two parameters /lambda_{N} and /lambda_{P} to weight the preference of beetles for parasitized ants due to their reduced aggressiveness.
+//Assumptions//
+  - Many many many...
+//Model//
+{{ :2017:groups:g4:model_advanced.jpg?nolink |}}
+//Preference//
+Consider the functional response between beetles and parasitized ants.
+{{ :2017:groups:g4:functional_beetles_parasitized.jpg?nolink |}}
+Let me tell you why this might not be such a crazy idea. The paper reports that there is a constant removal of parasitized ants by the beetles, independent of the beetles population. This functional response suggests this behaviour when beetles are close to their carrying capacity, and the preference towards parasitized ants is very high.
 Here we will assume that we are dealing with a sufficiently small beetles population for interference with each other to be negligible.
@@ Line 74: / Line 120: @@
 ===== Existence of equilibria and stability analysis =====
+Our model has three equilibrium points. The trivial equilibrium (0,0,0), the semi-trivial equilibirum (K,0,0) and a coexistence equilibrium with the following form:
+As it happens, there is an inequality which shows the transition of stability of the semi-trivial equilibrium by the appeareance of the coexistence equilibrium in the positive octant.
 ===== Numerical Simulations =====
@@ Line 159: / Line 209: @@
 //Functional Response//: the function relating the number of prey eaten by a single "average" predator to the size of the prey population.
-===== Questions =====
-Do parasitized ants go back to the colony?
-===== References =====
-{{ https://s-media-cache-ak0.pinimg.com/236x/c2/a1/eb/c2a1eb3d2e5a2b5de83d1b2c11528b48.jpg}}
-  * Mathis, K. A., and N. D. Tsutsui 2016 Dead ant walking: a myrmecophilous beetle predator uses parasitoid host location cues to selectively prey on parasitized ants. Proceedings of the Royal Society of London B 283:2016 1281
-  * Video of phorid flies in action: https://www.youtube.com/watch?v=pFVvOo1Qd_8
 ===== Reading List =====
@@ Line 171: / Line 215: @@
 Switching, Functional Response, and Stability in Predator-Prey Systems.
+===== Group presentantion =====
+  * [[http://200.145.112.249/webcast/files/Group%20IV_%20The%20Walking%20Dead%20Ant%20Tale-2.pdf|presentation]]