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This Concept Map, created with IHMC CmapTools, has information related to: c4. Logostic Equation, The tent map ???? The tent map is an iterated function, in the shape of a tent, forming a discrete-time dynamical system. It takes a point xn on the real line and maps it to another point:, To account for the many other variables that are not recorded, add (1-n) An+1 = rAn(1 − An) or in functional form ƒ(x) = rx (1 − x). ???? The intrinsic growth rate [or possibly the temperature gradient] (u) and the initial population size (xo) allow calculation of the (theoretical) size of the population in the subsequent generation (x+1)., c4. Logistic Equation ???? In 1845 Verholst described the growth of populations under limiting environmental conditions using a formula called the logistic equation., To account for the many other variables that are not recorded, add (1-n) An+1 = rAn(1 − An) or in functional form ƒ(x) = rx (1 − x). ???? The behavior of successive generations can then be calculated using the logistic equation. The logistic equation is dx/dt = ux(1-x) were 0<xə., If "An" in the logistic equation is the number of animals this year and "An+1" is the number next year, then An+1 = rAn where "r" is the growth rate or fecundity, will approximate the evolution of the population. ???? To account for the many other variables that are not recorded, add (1-n) An+1 = rAn(1 − An) or in functional form ƒ(x) = rx (1 − x)., Three behaviors of the logistic equation ???? Fixed: The population approaches a stable value., c4. Logistic Equation ???? The logistic equation is a simple equation or formula for approximating the evolution (of an animal population) over time., Chaotic: The population will eventually visit every neighborhood in a subinterval of (0, 1). Nested among the points it does visit, there is a countably infinite set of fixed points and periodic points of every period. ???? For a nonlinear equation to produce chaos, the equation requires a switchback or humped curve, non-inevertability, and at least one unstable fixed point., The behavior of successive generations can then be calculated using the logistic equation. The logistic equation is dx/dt = ux(1-x) were 0<xə. ???? If we set the value of u to be 2.5, meaning that the population can grow to 2.5 its size (doubles) in one generation; and a term (1-x where 0<xə) signifying those influences which prevent the population from reaching it’s maximum size; we find that the (actual) size of the population (x) reaches the value of 0.6X (0.6 times its maximum size) regardless of the size of the initial population. This is called a point attractor., The tent map has in it a prohibition to unlimited growth, but the conditional character of the map – if x is less than 0.5, we have one dynamics, (expansion) and if x is greater than 0.5, we have another (contraction). ???? Depending on the value of μ, the tent map demonstrates a range of dynamical behaviour ranging from predictable to chaotic., The tent map ???? The tent map and the logistic map are topologically conjugate, and thus the behaviour of the two maps are in this sense identical under iteration, Three behaviors of the logistic equation ???? Chaotic: The population will eventually visit every neighborhood in a subinterval of (0, 1). Nested among the points it does visit, there is a countably infinite set of fixed points and periodic points of every period., As u increases to reach 3.3, the size of the population size oscillates between two different values (0.48 and 0.83). This called a limit cycle. When the value of u is greater than 3.5, the population size oscillates between four values. ???? With values of u greater then 3.57, the population size varies chaotically without any specific pattern, but with repeating trajectories within boundaries, resulting in chaotic or strange attractors., Fixed: The population approaches a stable value. ???? The single attracting fixed point bifurcates repeatedly and then becomes chaotic. Note also the windows, In 1845 Verholst described the growth of populations under limiting environmental conditions using a formula called the logistic equation. ???? The logistic equation is a simple equation or formula for approximating the evolution (of an animal population) over time., The logistic equation is parabolic like the quadratic mapping with ƒ(0) = ƒ(1) = 0 and a maximum of ¼r at ½. Varying the parameter changes the height of the parabola but leaves the width unchanged. ???? The best way to visualize the behavior of the orbits as a function of the growth rate is with a bifurcation diagram. Pick a convenient seed value, generate a large number of iterations, discard the first few and plot the rest as a function of the growth factor. For parameter values where the orbit is fixed, the bifurcation diagram will reduce to a single line; for periodic values, a series of lines; and for chaotic values, a gray wash of dots., If we set the value of u to be 2.5, meaning that the population can grow to 2.5 its size (doubles) in one generation; and a term (1-x where 0<xə) signifying those influences which prevent the population from reaching it’s maximum size; we find that the (actual) size of the population (x) reaches the value of 0.6X (0.6 times its maximum size) regardless of the size of the initial population. This is called a point attractor. ???? As u increases to reach 3.3, the size of the population size oscillates between two different values (0.48 and 0.83). This called a limit cycle. When the value of u is greater than 3.5, the population size oscillates between four values., In 1845 Verholst described the growth of populations under limiting environmental conditions using a formula called the logistic equation. ???? The logistic equation represents the consequence of two opposing influences; 1) the tendency of a population to grow exponentially (at an ever increasing rate), 2) the effect of rate-limiting factors such a food supply or crowding (or predators) which keep the population below maximum size., If "An" in the logistic equation is the number of animals this year and "An+1" is the number next year, then An+1 = rAn where "r" is the growth rate or fecundity, will approximate the evolution of the population. ???? The logistic equation is parabolic like the quadratic mapping with ƒ(0) = ƒ(1) = 0 and a maximum of ¼r at ½. Varying the parameter changes the height of the parabola but leaves the width unchanged., The logistic equation is a simple equation or formula for approximating the evolution (of an animal population) over time. ???? The system might be better described as a discrete difference equation rather than as a continuous differential equation