Download presentation
Presentation is loading. Please wait.
1
The exponential function is a general description of a random process, where the probability of a certain event is independent of time C 0 /2 C 1 /2 C 2 /2 Lecture 2 Some basic functions and their application in biology
2
C 14 has a half time of 5568 years. How long would it take until of one mol C 14 only one atom remains? One mol contains 6.0210*10 23 atoms
3
The age of fossilized organic matter can be determined by the C 14 -method of radioactive decay. The half-live of C 14 is 5568 years. The equilibrium content of C 14 in living plants is about 10 -6 ppm (parts per million). How old is a fossilized plant with a C 14 content of 1.5*10 -7 ppm?
4
The logarithmic function Log-series relative abundance distribution Species – area relation
5
The power or allometric function
6
The Sierpinski triangle 1. Start with a triangle, 2. shrink to 1/2 size, 3. make three copies 4. arrange the three copies in quadrants 2,3, and 4 5. goto (2). Self similar objects
10
Self similarity
11
How life makes a complex pattern
12
Scaling factor = 1 / unit of measurement = magnification a is called the normalization constant
13
Ruler length l What is the value of a? If our object is classical Euclidean d = 0 Both equations match if b = cs Now we consider an area. The area scales to the square of the ruler length Both equations match if b = cs 2
14
Euclidean dimension E=1 Euclidean dimension E=2 Euclidean dimension E=3. Euclidean dimension E=0 E+d takes always values between the actual Euclidean dimension and the next higher one. It is commonly termed the fractal dimension of an object. 0 < d < 1 An important class of fractal objects are self similar objects. We describe them by power functions.
15
X = 0.75 What are the relation between radius, volumen and surface in such a branching pattern? A branching pattern
16
Calculate the total leaf area of this fern We have to measure two leafs at different scale to get the scaling exponent of the area - length relationship Let the average length of the smallest leaflets be 1 cm and its area 3 cm 2. At the next higher scale leaflet length might be 10 cm and the respective area 35 cm 2. The whole fern is 1 m long.
17
How should population density scale to body weight?
18
Population density is proportional to available space and to available energy.
19
MD is proportional to total population biomass What is if z is about 0.75? Energy equivalence (Damuth’s) rule (for poikilotherms: equal biomass hypothesis)
20
Species - area relationship
21
The mean number of bee species per km 2 in Poland [312685 km 2 ] is 110, the total number of Polish bees is 463. Estimate the number of bees in the district of Kujaw-Pommern [17970 km 2 ]. The mean number of bird species in Poland is about 430, the total European [10500000 km 2 ] species number is about 800. How many species do you expect for France [543965 km 2 ]? Would it make sense to estimate the species number of Luxembourg [2586 km 2 ]? What about Kujaw-Pommern [17970 km 2 ]? Observed: 530 Observed: 250 Observed: 262 Observed: unknown
22
The inverse hyperbola Michaelis-Menten equation Monod function
23
Haemoglobin or myoglobin bind oxygen according to the partial pressure of O2 Denoting y for [MbO], p(O2) for the partial pressure of oxygen and using [MbO] + [Mb] = const we get Hill equation of oxygen binding
24
Home work and literature Refresh: Fractal geometry Self similarity Branching processes Logarithmic transformations Species – area relationships Radioactive decay Prepare to the next lecture: Vectors Vector operations (sum, S-product, scalar product) Scalar product of orthogonal vectors Distance metrics (Euclidean, Manhattan, Minkowski) Cartesian system, orthogonal vectors Matrix Types of matrices Basic matrix operations (sum, S- product, dot product) Literature: Mathe-online Fractal geometry: http://classes.yale.edu/fractals/ http://classes.yale.edu/fractals/ Fractals: http://en.wikipedia.org/wiki/Fractalhttp://en.wikipedia.org/wiki/Fractal
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.