The Spotted Owls 5 " 5. = 5 " œ!þ")45 Ð'!% leave nest, 30% of those succeed) + 5 " œ!þ("= 5!Þ*%+ 5 ( adults live about 20 yrs)

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Mervyn Harrington
5 years ago
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1 The Spotted Owls Be sure to read the example at the introduction to Chapter in the textbook. It gives more general background information for this example. The example leads to a dynamical system which we can understand better using complex eigenvalues. The life cycle of the spotted owl is divided into 3 stages in this model: j s a juvenile subadult" adult The model, as described, assumes there is an approximately 1:1 ratio between male and female owls and counts only the females. There is a stage matrix E that relates the number of owls in each group to the number a year later. The matrix, with entries based on field data, is so that Ô 4 " Ô!! Þ$$ Ô 4 = " œ Þ")!! = Õ+ Õ! Þ(" Þ*% Õ+ " B " œ E B Ú4 œ number of juvenile spotted owls at year where Û= œ number of subadult spotted owls at year Ü+ œ number of adult spotted owls at year 4 " œ!þ$$+ = " œ!þ")4 Ð'!% leave nest, 30% of those succeed) + " œ!þ("=!þ*%+ ( adults live about 20 yrs) From Matlab, the eigenvectors are the ß@ ß@ " # $ of the matrix T œ Ô!Þ!)%)!Þ'(')3!Þ!)%)!Þ'(')3!Þ$"(&!Þ&*#)!Þ!""%3!Þ&*#)!Þ!""%3!Þ!&)" Õ!Þ%"'(!Þ!*('3!Þ%"'(!Þ!*('3!Þ*%'& and the corresponding eigenvalues are - " œ ! # œ $ œ!þ*)$'

2 Of course, a nonzero multiple of an is still an eigenvector with the same eigenvalue, because a multiple is in the same eigenspace itself. Therefore Ô!Þ$"(& Ô $"(& 10000*!Þ!&)" œ &)" is also an eigenvector with eigenvalue - $ Õ!Þ*%'& To get rid of some decimals, we will use this rescaled vector as a replacement for the eigenvector in column 3 of the matrix T. Ô $"(& 1) If the owl population begins (say) as B! $ œ &)" ß then the populations in successive years are and in general, B" œ EB! œ - $ B! B# œ EÐB" Ñ œ EÐEB! Ñ œ EÐ-$ B! Ñ œ -$ EÐB! Ñ œ -$ # B! B œ EÐB Ñ œ EÐEÐEB Ñ œ EE # ÐB Ñ œ EÐ-$ # B Ñ œ -$ # EB œ -$ $ B ã B œ EÐB Ñ œ E ÐB Ñ œ ÞÞÞ œ - B $ #!!!!! "! $! Ð!Þ*)$'Ñ Ô $"(&! &)" Ä Ô 8! 8 Õ! as Ô $"(& Therefore: for an initial owl population B! œ &)", the model predicts that the population will eventually die out. 2) Suppose the owl population begins at B!. Since we are thinking of B! as a population vector, we assume its entries are nonnegative real numbers Ðbut from a purely mathematical point of view, all the following calculations work even if B! has negative or any complex entriesñ. T has $ distinct eigenvalues, and the corresponding ß@ # ß@ $ are a basis for $. Note that we could also check that these vectors are a basis by confirming (via Matlab) that T is invertible. Ô $"(& B! œ -"-"@" -# -#@# -$ Ð!Þ*)$'Ñ &)" In this expression the weights -3 might be any complex numbers Þ But B! is a real vector (it was a given population vector) so whatever the - 3 's are, the imaginary parts of all the entries will cancel out to produce the real vector B! Þ

3 And, since E is a real matrix, B œ EB ß B œ EB ß... are also all real. "! # " Just as in the first example, multiplying 8 times by E leads to the equation B œ Ð!Þ*)$'Ñ " " " # # # $ Ô $"(& &)" È # # Recall that if -= +,3 is any complex number, then l-l œ +, is the distance from - to the origin in the complex plane. In the case of the eigenvalue -, l- l œ ÈÐ!Þ!#")Ñ# Ð!Þ#!&*Ñ#!Þ#!(" "Þ " " Therefore as 8 Ä, l-" l œ l-" l Ð!Þ#!("Ñ Ä!. That is, -" gets closer and closer to the the origin: - 8 " Ä! Similarly, as 8 Ä, l- l œ l- l Ð!Þ#!("Ñ Ä!ß so - Ä!. # # # So B 8 is a sum of three terms each of which tends to! as 8 Ä. So once again we conclude again that B 8 Ä! : for the given stage matrix E, the spotted owl population will eventually die out; it doesn't matter what the initial population B! is. In this dynamical system with the given matrix E, the origin! is called a universal attractor that is, B 8 Ä! for every choice of initial population vector B!. More informally, no matter where the system starts, the origin attracts the B 8 's as the dynamical system progresses. As the textbook states, the most crucial number in the model is the 0.18 in the original matrix E À the percentage of juvenile owls who (by finding a territory of their own and a mate) manage to pass into the subadult stage. Their success depends, in part, on how much good territory is available for the owls. If the success rate 0.18 for juveniles passing into the subadult stage is improved to (say)!! Þ$$!Þ&!, then the matrix for the dynamical system becomes Ô Þ&!! Þ Õ! Þ(" Þ*% The eigenvalues and corresponding Ðlinearly independent Ñ eigenvectors turn out to be Ô!Þ#!*!!Þ%$(!3 - " œ 0.!&$% 0. " œ!þ&*%)!þ%"#)3 Õ!Þ%(""!Þ"$)%3 Ô!Þ#!*!!Þ%$(!3 -# œ -" œ 0.!&$% 0. # œ!þ&*%)!þ%"#)3 Õ!Þ%(""!Þ"$)%3

4 - $ œ "Þ!%'* Ô $ œ "%#" If we write the initial population vector as a linear combination of these, B! B œ Ð"Þ!%'*Ñ! " " " # # # $ 8 Ô #*(' "%#" (*) Then B œ Ð"Þ!%'*Ñ " " " # # # $ Ô #*(' "%#" 8 8 As before, l- l and l- l are " so - and - Ä! as 8 Ä, " # " # 8 8 so - - Ä! Þ Therefore, when 8 is large " " " # # # #*(' B 8 - Ð"Þ!%'*Ñ Ô 8 $ "%#" ( exp growth*** ) (**) This means that if B! was chosen so that - $ Á 0, the population continues to grow. ( Of course, at some time the population would become so large that the model would break down Þ For example, there will eventually not be enough good territory available to maintain the 0% success rate for owls passing from the juvenile to the subadult stage.) Assume that there's enough territory and other resources to allow 8 to become big enough that the approximation in Ð Ñ is a fair statement. From that time on (until the model breaks down): i) each new population vector B 8 " is approximately "Þ!%'* times the preceding population vector: a growth rate of about %Þ(%/year. (***- $ controls long term growth) ii) the population vector remains, each year, (approximately) a constant multiple Ô #*(' of "%#". This means the population of juveniles, subadults and adults each stabilize at a fixed % of the total owl population: for example, the juveniles at #*(' Ð#*(' "%#" *%%"Ñ "!! #"Þ&% of the population.

5 To take a specific B! À suppose the matrix given above (with the 0% juvenile --> subadult rate) is accurate for a forested region with no spotted owls and that a population consisting Ô! entirely of "!!! adult owls is introduced: so B! œ! Õ" 000 In that case, from Matlab, the coefficients -" ß-# ß-$ that work in Ð Ñ, rounded to % places, are -" œ #Þ$%*% "Þ)("$3 -# œ #Þ$%*% "Þ)("$3 -$ œ!þ!!!* Some sample calculations give ( rounded to the nearest owl) Ô $$! Ô %"$ Ô %$$ B& œ "&' B"! œ "*) B"" œ #!( Õ"!%% Õ"$"$ Õ"$(% Å Å (Approximately what % growth?) The moral is that the eigenvalues and basic eigenvectors control the long term behavior of the system.

6 Another Example Suppose we have a linear dynamical system B " œ EB where E (for simplicity) is # # and where E is diagonalizable, with eigenvalues -" ß -# corresponding to eigenbasis ß@ # Þ! " " # # "! " " # # " " " # # # We can write the initial vector B œ so B œ EB œ - E@ - E@ œ - - # # B# œ EB " œ -"-" E@ " -# -# E@ # œ -"-"@" -# -#@# ã " " B œ EB œ œ - - E@ - - E@ œ - - " " " " # # # " " " # # # We can consider what happens depending on the size of the eigenvalues. To illustrate with one case: when - " and l- l " " # Write B! œ ã B œ "- #- " " # # If - œ! À Then B œ and B Ä! " #-# # If - Á 0 À Since - Ä! as Ä, we get that, for large, " # # # B - " " B - œ - B " " " " " " So, for large, B will grow approximately by a factor of -" each time so -" controls the long term growth rate of the system. Example: Predator and prey: owl and wood rat population in a forest (see Example 1, p. 302 in the textbook) S œ number of owls ß V œ number of rats (measured in rat-units = 1000's) Model: S " œ Þ&S Þ%V V œ :S "Þ"V " B S " Þ& Þ% S Þ& Þ% œ œ œ B V : "Þ" V : "Þ" " "

7 1) For : œ!þ#!&, the eigenvalues are approximately.7106 and What happens? 2) For : œ!þ08ß the eigenvalues are approximately - œ and - œ.92, with " # Þ&*%' Þ*)*# corresponding œ œ Þ)!%! Þ"%'$ Þ So, for large B œ ( since - # Ä!) " " # # " " B œ S - Ð"Þ!%!)Ñ Þ&*%' V " Þ)!%! ( The value of the constant - ", of course, is determined by the initial population B!.) This says that for large, the ratio of owls to rats œ O V -" Ð"Þ!%!)Ñ ÐÞ&*%'Ñ - Ð"Þ!%!)ÑÐÞ)!%!Ñ " œ Þ&*%' Þ)!%! Þ(% $ % Ð3 owls / 4000 rats) S+1 S Also, Ð"Þ!%!)Ñ so for large Ðand as long as the model V V +1 holds up Ñ both populations grow at a rate of about 4.08%/month. = Rats Slope 4/3 rather than 3/4 here, only because I used the horizontal axis á for the owl-axis. Owls

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