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"Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellID->746232340], Cell[CellGroupData[{ Cell["CAPTION", "Section", CellFrame->{{0, 0}, {1, 0}}, CellFrameColor->RGBColor[0.87, 0.87, 0.87], FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.597406, 0, 0.0527047]], Cell["\<\ This population dynamics model (being the extension of the well-known Levins \ metapopulation model) describes the competition for territory of two species \ having different colonization strategies. The model equations are:\ \>", "Text"] }, Close]] }, Open ]], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["p", "1"], "'"}], RowBox[{"(", "t", ")"}]}], " ", "=", RowBox[{ RowBox[{ SubscriptBox["c", "1"], RowBox[{ SubscriptBox["p", "1"], "(", "t", ")"}], RowBox[{"(", RowBox[{"1", "-", RowBox[{ SubscriptBox["p", "1"], "(", "t", ")"}], "-", RowBox[{ SubscriptBox["p", "2"], "(", "t", ")"}]}], ")"}]}], "-", RowBox[{ SubscriptBox["e", "1"], RowBox[{ SubscriptBox["p", "1"], "(", "t", ")"}]}], " ", "+", " ", RowBox[{"o", " ", RowBox[{ SubscriptBox["p", "1"], "(", "t", ")"}], RowBox[{ SubscriptBox["p", "2"], "(", "t", ")"}]}]}]}], TraditionalForm]], "InlineMath"], ",", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["p", "2"], "'"}], RowBox[{"(", "t", ")"}]}], " ", "=", " ", RowBox[{ RowBox[{ SubscriptBox["c", RowBox[{"2", " "}]], RowBox[{ SubscriptBox["p", "2"], "(", "t", ")"}], RowBox[{"(", RowBox[{"1", "-", RowBox[{ SubscriptBox["p", "1"], "(", "t", ")"}], "-", RowBox[{ SubscriptBox["p", "2"], "(", "t", ")"}]}], ")"}]}], "-", RowBox[{ SubscriptBox["e", "2"], " ", RowBox[{ SubscriptBox["p", "2"], "(", "t", ")"}]}], " ", "-", " ", RowBox[{"o", " ", RowBox[{ SubscriptBox["p", "1"], "(", "t", ")"}], RowBox[{ SubscriptBox["p", "2"], "(", "t", ")"}]}]}]}], TraditionalForm]], "InlineMath"], "," }], "Text"], Cell[TextData[{ "where ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "1"], "(", "t", ")"}], TraditionalForm]], "InlineMath"], " and ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "2"], "(", "t", ")"}], TraditionalForm]], "InlineMath"], " are the portion of the territory at time ", Cell[BoxData[ FormBox["t", TraditionalForm]], "InlineMath"], " occupied by species 1 and 2, respectively. The occupation is exclusive. \ Hence, the assumptions ", Cell[BoxData[ FormBox[ RowBox[{ RowBox[{"0", "\[LessEqual]", RowBox[{ SubscriptBox["p", "1"], "(", "t", ")"}]}], ",", RowBox[{"0", "\[LessEqual]", RowBox[{ SubscriptBox["p", "2"], "(", "t", ")"}]}]}], TraditionalForm]], "InlineMath"], " and ", Cell[BoxData[ FormBox[Cell[TextData[Cell[BoxData[ FormBox[ RowBox[{ RowBox[{ RowBox[{ SubscriptBox["p", "1"], "(", "t", ")"}], "+", RowBox[{ SubscriptBox["p", "2"], "(", "t", ")"}]}], "\[LessEqual]", "1"}], TraditionalForm]]]], "InlineMath"], TraditionalForm]]], " hold. " }], "Text"], Cell["The meaning of the parameters is as follows:", "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ SubscriptBox["c", "1"], TraditionalForm]], "InlineMath"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["c", "2"], "\[GreaterEqual]", "0"}], TraditionalForm]], "InlineMath"], ", the global rates of colonization, describe how the species can colonize \ empty patches. The global rates of extinction are ", Cell[BoxData[ FormBox[ SubscriptBox["e", "1"], TraditionalForm]], "InlineMath"], ", ", Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["e", "2"], "\[GreaterEqual]", "0"}], TraditionalForm]], "InlineMath"], ". The global rate of relative overcolonization ", Cell[BoxData[ FormBox["o", TraditionalForm]], "InlineMath"], " describes how the species can occupy the patches occupied by the other \ species. " }], "Text"], Cell[CellGroupData[{ Cell["DETAILS", "Section", CellFrame->{{0, 0}, {1, 0}}, CellFrameColor->RGBColor[0.87, 0.87, 0.87], FontFamily->"Helvetica", FontSize->12, FontWeight->"Bold", FontColor->RGBColor[0.597406, 0, 0.0527047]], Cell["\<\ In general, overcolonization is the phenomenon of a certain species occupying \ territories already being occupied by other species. In the case of \ preemptive competition there is no interaction between the species; they can \ occupy only empty patches. In the case of hierarchical relationship, species \ 1 can occupy both empty patches and those that are occupied by species 2. \ Species 2 can colonize only empty patches. In general cases of \ overcolonization both species can colonize both empty and occupied patches. \ \>", "Text"], Cell["In our model, these cases are represented as follows:", "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", RowBox[{"1", " "}]], "\[ReverseUpEquilibrium]", " ", SubscriptBox["p", "2"]}], TraditionalForm]], "InlineMath"], ": ", Cell[BoxData[ FormBox[ RowBox[{"o", "=", "0"}], TraditionalForm]], "InlineMath"], "\[LongDash]no overcolonization, preemptive model." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "1"], "\[LongRightArrow]", SubscriptBox["p", "2"]}], TraditionalForm]], "InlineMath"], ": ", Cell[BoxData[ FormBox[ RowBox[{"o", "=", SubscriptBox["k", "1"]}], TraditionalForm]], "InlineMath"], "\[LongDash]species 1 can occupy the empty territories of species 2." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ SubscriptBox["p", "2"], "\[LongRightArrow]", SubscriptBox["p", "1"]}], TraditionalForm]], "InlineMath"], ": ", Cell[BoxData[ FormBox[ RowBox[{"o", "=", RowBox[{"-", SubscriptBox["k", "2"]}]}], TraditionalForm]], "InlineMath"], "\[LongDash]species 2 can occupy the empty territories of species 1." }], "Text"], Cell[TextData[{ Cell[BoxData[ FormBox[ RowBox[{ SubsuperscriptBox[ SubscriptBox["p", RowBox[{"1", " "}]], "\[LongLeftArrow]", "\[LongRightArrow]"], " ", SubscriptBox["p", "2"]}], TraditionalForm]], "InlineMath"], ": ", Cell[BoxData[ FormBox[ RowBox[{"o", ">", "0"}], TraditionalForm]], "InlineMath"], "\[LongDash]species 1 is stronger; it can occupy more of species 2's \ territories than species 2 can occupy of species 1's territories. ", Cell[BoxData[ FormBox[ RowBox[{"o", "<", "0"}], TraditionalForm]], "InlineMath"], " gives the opposite case." }], "Text"], Cell["\<\ Besides the trivial equilibrium, there are always two more equilibria, where \ one of the variables is zero and the other is positive. One can observe that \ no coexistence of species is possible if no interaction is allowed. The \ positive asymptotically stable equilibrium can exist only if there is some \ kind of interaction between the species.\ \>", "Text"], Cell[TextData[{ "The equilibria of the preemptive competition are (0, 0), ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"1", "-", FormBox[ FractionBox[ SubscriptBox["e", "1"], SubscriptBox["c", "1"]], TraditionalForm]}], ",", "0"}], ")"}], TraditionalForm]], "InlineMath"], " and ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"0", ",", RowBox[{"1", "-", FormBox[ FractionBox[ SubscriptBox["e", "2"], SubscriptBox["c", "2"]], TraditionalForm]}]}], ")"}], TraditionalForm]], "InlineMath"], "." }], "Text"], Cell[TextData[{ "The equilibria of the hierarchical overcolonization model are (0, 0), ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"1", "-", FormBox[ FractionBox[ SubscriptBox["e", "1"], SubscriptBox["c", "1"]], TraditionalForm]}], ",", "0"}], ")"}], TraditionalForm]], "InlineMath"], ", ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"0", ",", RowBox[{"1", "-", FormBox[ FractionBox[ SubscriptBox["e", "2"], SubscriptBox["c", "2"]], TraditionalForm]}]}], ")"}], TraditionalForm]], "InlineMath"], " and ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ RowBox[{"-", RowBox[{ SubscriptBox["c", "1"], "(", RowBox[{ SubscriptBox["e", "2"], "+", SubscriptBox["c", "1"]}], ")"}]}], "+", RowBox[{ SubscriptBox["e", "1"], "(", RowBox[{ SubscriptBox["c", "1"], "+", SubscriptBox["c", "2"]}], ")"}]}], RowBox[{ SubscriptBox["c", "1"], SubscriptBox["c", "2"]}]], ",", RowBox[{"1", "-", FractionBox[ SubscriptBox["e", "1"], SubscriptBox["c", "1"]]}]}], ")"}], TraditionalForm]], "InlineMath"], "." }], "Text"], Cell[TextData[{ "The equilibria of the general overcolonization model are (0, 0), ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ RowBox[{"1", "-", FormBox[ FractionBox[ SubscriptBox["e", "1"], SubscriptBox["c", "1"]], TraditionalForm]}], ",", "0"}], ")"}], TraditionalForm]], "InlineMath"], ", ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{"0", ",", RowBox[{"1", "-", FormBox[ FractionBox[ SubscriptBox["e", "2"], SubscriptBox["c", "2"]], TraditionalForm]}]}], ")"}], TraditionalForm]], "InlineMath"], " and ", Cell[BoxData[ FormBox[ RowBox[{"(", RowBox[{ FractionBox[ RowBox[{ RowBox[{ SubscriptBox["e", "2"], SubscriptBox["c", "1"]}], "-", RowBox[{ SubscriptBox["e", "1"], SubscriptBox["c", "2"]}], "+", " ", RowBox[{"o", " ", RowBox[{"(", RowBox[{ RowBox[{"-", SubscriptBox["e", "2"]}], "+", SubscriptBox["c", "2"]}], ")"}]}]}], RowBox[{"o", "(", RowBox[{"o", "-", SubscriptBox["c", "1"], "+", SubscriptBox["c", "2"]}], ")"}]], ",", " ", FractionBox[ RowBox[{ RowBox[{"c", " ", RowBox[{"(", RowBox[{ SubscriptBox["e", "1"], "-", SubscriptBox["c", "1"]}], ")"}]}], "-", RowBox[{ SubscriptBox["e", "2"], SubscriptBox["c", "1"]}], "+", RowBox[{ SubscriptBox["e", "1"], SubscriptBox["c", "2"]}]}], RowBox[{"o", "(", RowBox[{"o", "-", SubscriptBox["c", "1"], "+", SubscriptBox["c", "2"]}], ")"}]]}], ")"}], TraditionalForm]], "InlineMath"], "." }], "Text"], Cell["\<\ In practice, weeds are very aggressive in colonization and their extinction \ can be very low. Other species like pine or oak colonies can colonize slowly \ but their extinction is also low.\ \>", "Text"], Cell["References: ", "Text"], Cell[TextData[{ "I. Szimjanovszki, \"Computer-Aided Analysis of the Growth of the \ Territory-Occupational Populations Using ", StyleBox["Mathematica", FontSlant->"Italic"], ",\" master's thesis, 2008." }], "Text"], Cell[TextData[{ "E. V. P. Racz and J. Karsai, \"Computer Simulation Results for Cellular \ Automata Models of Some Ecological Systems,\" ", StyleBox["Folia FSN Universitatis Masarykianae Brunensis, Mathematica", FontSlant->"Italic"], ", ", StyleBox["13", FontWeight->"Bold"], ", 2003b pp. 213\[Dash]221." }], "Text"] }, Close]], Cell[CellGroupData[{ Cell["THIS NOTEBOOK IS THE SOURCE CODE FROM", "Text", CellFrame->{{0, 0}, {0, 0}}, CellMargins->{{48, 10}, {4, 28}}, CellGroupingRules->{"SectionGrouping", 25}, CellFrameMargins->{{48, 48}, {6, 5}}, CellFrameColor->RGBColor[0.87, 0.87, 0.87], FontFamily->"Helvetica", FontSize->10, FontWeight->"Bold", FontColor->RGBColor[0.597406, 0, 0.0527047]], Cell[TextData[{ "\"", ButtonBox["Competition for Territory: The Levins Model for Two Species", BaseStyle->"Hyperlink", ButtonData->{ URL["http://demonstrations.wolfram.com/\ CompetitionForTerritoryTheLevinsModelForTwoSpecies/"], None}, ButtonNote-> "http://demonstrations.wolfram.com/\ CompetitionForTerritoryTheLevinsModelForTwoSpecies/"], 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