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6.3: Biological-ecological graph - Biology

6.3: Biological-ecological graph - Biology


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Figure (PageIndex{1}) plots the two green columns of Table 6.1.1 through line 12—the mid-1960s—in blue dots, with a green line representing the average trend. This one was done using a statistical “regression” program, with r the point at which the line intersects the vertical axis and s the line’s slope— its ∆y /∆x. The intrinsic growth rate r for modern, global human population is apparently negative and the slope s is unmistakably positive.

From the late 1600s to the mid 1960s, then, it’s clear that the birth rate per family was increasing as the population increased. Greater population was enhancing the population’s growth. Such growth is orthologistic, meaning that the human population has been heading for a singularity for many centuries. The singularity is not a modern phenomenon, and could conceivably have been known before the 20th century.

The negative value of r, if it is real, means there is a human Allee point. If the population were to drop below the level of the intersection with the horizontal axis—in this projection, around two hundred million people—the human growth rate would be negative and human populations would decline. The Allee point demonstrates our reliance on a modern society; it suggests that we couldn’t survive with our modern systems at low population levels—although perhaps if we went back to hunter–gatherer lifestyles, this would change the growth curve. The Allee point thus indicates that there is a minimum human population we must sustain to avoid extinction. We depend on each other.


HIV and AIDS

Case Study: Newborn baby ‘cured’ of HIV with rapid detection and ARV treatment. Great introduction to the topic from the New York Times.

Elizabeth Pisani discusses rational behaviour in HIV infection, with lots of Indonesia references:

90-day time-lapse of a woman on modern anti-AIDS medication:

Could the design of a condom improve HIV prevention? Here is a South African company who hope so: Pronto Condoms. For a video of how it works (on a plastic model), click here.

Mosaic Science Magazine talks to Françoise Barré-Sinoussi about how she identified HIV as the cause of AIDS, her receipt of the Nobel Prize, and the latest efforts to prevent, treat and manage HIV. [Photo Credit: CC-BY: Ben Gilbert/Wellcome Images]. Click to read.


6.3 Volumes of Revolution: Cylindrical Shells

In this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution. We can use this method on the same kinds of solids as the disk method or the washer method however, with the disk and washer methods, we integrate along the coordinate axis parallel to the axis of revolution. With the method of cylindrical shells, we integrate along the coordinate axis perpendicular to the axis of revolution. The ability to choose which variable of integration we want to use can be a significant advantage with more complicated functions. Also, the specific geometry of the solid sometimes makes the method of using cylindrical shells more appealing than using the washer method. In the last part of this section, we review all the methods for finding volume that we have studied and lay out some guidelines to help you determine which method to use in a given situation.

The Method of Cylindrical Shells

Again, we are working with a solid of revolution. As before, we define a region R , R , bounded above by the graph of a function y = f ( x ) , y = f ( x ) , below by the x -axis, x -axis, and on the left and right by the lines x = a x = a and x = b , x = b , respectively, as shown in Figure 6.25(a). We then revolve this region around the y-axis, as shown in Figure 6.25(b). Note that this is different from what we have done before. Previously, regions defined in terms of functions of x x were revolved around the x -axis x -axis or a line parallel to it.

To calculate the volume of this shell, consider Figure 6.27.

The shell is a cylinder, so its volume is the cross-sectional area multiplied by the height of the cylinder. The cross-sections are annuli (ring-shaped regions—essentially, circles with a hole in the center), with outer radius x i x i and inner radius x i − 1 . x i − 1 . Thus, the cross-sectional area is π x i 2 − π x i − 1 2 . π x i 2 − π x i − 1 2 . The height of the cylinder is f ( x i * ) . f ( x i * ) . Then the volume of the shell is

Note that x i − x i − 1 = Δ x , x i − x i − 1 = Δ x , so we have

Another way to think of this is to think of making a vertical cut in the shell and then opening it up to form a flat plate (Figure 6.28).

In reality, the outer radius of the shell is greater than the inner radius, and hence the back edge of the plate would be slightly longer than the front edge of the plate. However, we can approximate the flattened shell by a flat plate of height f ( x i * ) , f ( x i * ) , width 2 π x i * , 2 π x i * , and thickness Δ x Δ x (Figure 6.28). The volume of the shell, then, is approximately the volume of the flat plate. Multiplying the height, width, and depth of the plate, we get

which is the same formula we had before.

To calculate the volume of the entire solid, we then add the volumes of all the shells and obtain

Here we have another Riemann sum, this time for the function 2 π x f ( x ) . 2 π x f ( x ) . Taking the limit as n → ∞ n → ∞ gives us


6.3.7 Outline the effects of HIV on the immune system.

The HIV virus (which causes AIDS) destroys a type of lymphocyte which has a vital role in antibody production. Over the years this results in a reduced amount of active lymphocytes. Therefore, less antibodies are produced which makes the body very vulnerable to pathogens. A pathogen that could easily be controlled by the body in a healthy individual can cause serious consequences and eventually lead to death for patients affected by HIV.


General Network Properties

As degree degi, we define the total number of edges adjacent to a vertex. In the case of a directed graph we distinguish between the “indegree” ( d e g i i n ) and “outdegree” ( d e g i o u t ) . The indegree refers to the number of arcs, incident from the vertex, whereas the outdegree to the number of arcs incident to the vertex. In a social network for example, the indegree would represent the followers, whereas the outdegree the people one follows. The total degree in a directed graph is the sum of the indegree and outdegree d e g i = d e g i i n + d e g i o u t showing all connections (both followers and followed people). The average degree of the network is d e g a v g = Σ d e g i V (Figure 3A). Looking at all nodes in a network, in order to study the degree distribution p(k), we consider the probability that a randomly selected vertex has degree equal to k. The same information can also be found as cumulative degree distribution pc(k) which shows the a-posterior probability of a randomly selected vertex to have degree larger than k. Notably, the degree distribution is one of the most important topological features and is characteristic to different network types. In the simplest case, p(k) can be estimated by a histogram of degrees. An example is shown in Figure 3B. Networks, whose degree distribution follow a power law, are called scale-free networks.

Figure 3. Network properties and topological features. (A) A network G = (V, E) consisting of V = 18 nodes and E = 21 edges. Each node's size has been adjusted according to its degree. Vertex V1 for example has 10 neighbors, thus degree d(V1) = 10. The average degree for the whole network is 42 18 = 2 . 333 . Network has been visualized with Cytoscape. (B) A scatterplot histogram showing the degree distribution. The Y axis holds the values about how many nodes have certain degree (values in X axis). (C) Clustering coefficient. Node V has 6 neighbors <V1 V2, V3, V4, V5, V6>. The maximum number of edges between these neighbors are 6 ( 6 - 1 ) 2 = 15 but only two neighbors (V1 and V2) are connected to each other thus making the clustering coefficient for node V equal to 1 15 = 0 . 066 . (D) Similarly, the neighbors of node V are connected with 11 edges between each other (E = <<>1,V2>, <>1,V5>, <>1,V4>, <>2,V3>, <>2,V6>, <>2,V4>, <>3,V6>, <>3,V5>, <>4,V6>, <>4,V5>, <>5,V6>>), the clustering coefficient for this node will be C V = 11 15 =   0.733. Notably dotted lines represent the direct connections of node V, whereas the solid lines represent the connections between the first neighbors of node V. (E) The closeness centrality in blue, the betweenness centrality in red and the eccentricity centrality in orange. The graph consists of 6 nodes and 5 edges. Closeness centrality calculation example: Node V1accesses nodes V2, V4, V5, V6 with step 1 and node V3 with step 2. Therefore, its closeness centrality is calculated as 5 4 × 1 + 2 × 1 = 5 6 = 0 . 833 . Betweenness centrality calculation example: Since all nodes are accessible through any other node, there are N(N − 1) = 6 × 5 = 30 shortest paths but only 12 of them pass through node V2. These are <V3, V2>, <V3, V2, V1>,<V3, V2, V1, V4>, <V3, V2, V1, V5>, <V3, V2, V1, V6>, <V2, V1>, <V2, V1, V4>, <V2, V1, V6>, <V2, V1, V5>, <V4, V1, V2, V3>, <V5, V1, V2, V3> and <V6, V1, V2, V3>. Therefore the C b e t ( V 2 ) = 12 30 = 0 . 4 . Eccentricity calculation example: Node V1 accesses nodes V2, V4, V5, V6 with one step and node V3 with two steps. Therefore, its eccentricity will be max (2, 1) = 2.

Density is the ratio between the number of edges in a graph and the number of possible edges in the same graph. In a fully connected graph (e.g., protein complex), the number of possible edges (pairwise connections) are E m a x = V ( V - 1 ) 2 . Therefore, the density can be calculated as d e n s i t y = E E m a x = 2 E V ( V - 1 ) . If a graph has EV k , 2 > k > 1, then this graph is considered as dense, whereas when a graph has EV or EV k , k ≤ 1, it is considered as sparse.

The Clustering coefficient is a measure which shows whether a network or a node has the tendency to form clusters or tightly connected communities (e.g., protein clusters in a protein-protein interaction network). The clustering coefficient of a node is defined as the number of edges between its neighbors divided by the number of possible connections between these neighbors. The clustering coefficient of a node i is defined as C i = 2 e k ( k - 1 ) where k is the number of neighbors (degree) and e the number of edges between these k neighbors. The average clustering of a network is given by C a v g = Σ C i V . The clustering coefficient takes values 0 ≤ Ci ≤ 1, thus the closer to 1, the higher the tendency for clusters to be formed. An example is shown in Figures 3C,D.

The matching index Mij can be used to identify two nodes in a network which might be functionally similar without necessarily being connected to each other. The matching index is a measure to quantify such similarity between any two nodes within a network and, according to the above, two nodes can be found to be functionally similar if they share common neighbors. The matching index between vertices i and j is calculated as M i j = Σ   d i s t i n c t   c o m m o n   n e i g h b o r s Σ   t o t a l   n u m b e r   o f   n e i g h b o r s and can be extended beyond the direct neighbors of a vertex. In addition, it can be applied to multi-edge networks.

The distance distij between two nodes (e.g., metabolites in a metabolic network) is defined as the length of the shortest path between them. As shortest path we define the minimal number of edges that need to be traversed to reach node j from node i. In the case where two shortest paths of identical length exist, any of them could be used. Whenever there is no connection between two nodes i and j, then their distance is defined as infinite distij = ∞. In addition, the diameter, diamm = max(distij), is the maximal distance between any pair of vertices. The average path length is defined as the average distance between all node pairs and is defined as d i s t a v g = 1 N ( N - 1 ) ∑ i   =   1 N ∑ j   =   1 N d i s t i j .


Resources

SAT and SAT Subject Tests Student Registration Booklet

Includes registration requirements and instructions, test schedules, and contact information.

Flyer: How to Link Your Accounts

Step-by-step guide on how to link College Board and Khan Academy accounts for a personalized SAT practice plan.

College Application Fee Waiver Directory of Colleges

List of colleges and universities cooperating with the College Application Fee Waiver program.

Test Specifications for the Redesigned SAT

Full specifications for the redesigned SAT. Includes evidentiary foundations for redesign, as well as sample questions.


Biological Sciences (Conservation Biology and Ecology) (BS) Accelerated Program

Ecology is the study of the distribution and abundance of organisms, the interactions among organisms, and the interactions between organisms and the physical environment. Conservation biology is an applied science based on ecological principles that focuses on conserving biological diversity and on restoring degraded ecosystems.

Arizona State University is committed to a more sustainable world and sharing knowledge of conservation biology and ecology through the BS program in biological sciences with a concentration in conservation biology and ecology is one critical component to help meet this global challenge.

Conservation biologists at ASU investigate the impact of humans on Earth's biodiversity and develop practical approaches to prevent the extinction of species and promote the sustainable use of biological resources. Some investigate the causes of ecosystem degradation and use ecological principles to reestablish desired conditions in a variety of ecosystems, including rivers, wetlands, grasslands, urban landscapes and forests.

Due to the high volume of overlap in curriculum, students enrolled in this degree are not permitted to declare a concurrent degree combination with any other program within the School of Life Sciences. Students should speak with their academic advisor for any further questions.


6.3: Biological-ecological graph - Biology

To understand linear relationships in biology, we must first learn about linear functions and how they differ from nonlinear functions.

Definition: Linear and Nonlinear Functions

The key feature of linear functions is that the dependent variable (y) changes at a constant rate with the independent variable (x). In other words, for some fixed change in x there is a corresponding fixed change in y. As the name implies, linear functions are graphically represented by lines.

Definition: A linear function is a function that has a constant rate of change and can be represented by the equation y = mx + b, where m and b are constants. That is, for a fixed change in the independent variable there is a corresponding fixed change in the dependent variable.

If we take the change in x to be a one unit increase (e.g., from x to x + 1), then a linear function will have a corresponding constant change in the variable y. This idea will be explored more in the next section when slope is discussed.

Nonlinear functions, on the other hand, have different changes in y for a fixed change in x.

Definition: A nonlinear function is a function that is not linear. That is, for a fixed change in the independent variable, there is NOT a corresponding fixed change in the dependent variable.

The following graph depicts a nonlinear function with a non constant rate of change,

In this example, there is both a 5 unit increase in y and a 11 unit decrease in y corresponding to a one unit increase in x. A nonlinear function does not exhibit a constant rate of change, and therefore is not graphically represented by a line. In fact, you probably think of nonlinear functions as being curves. The following table summarizes some of the general differences between linear and nonlinear functions:

Linear Functions

Domain and range is all real numbers.

Graphically represented by a straight line.

Nonlinear Functions

Domain and range can vary.

Often graphically represented by a curve.

Representing linear functions

Linear functions can be written in slope-intercept form as,

We can use the slope-intercept form of a line to demonstrate that a linear function has a constant rate of change. To see this, consider a one unit increase in x (i.e. from x to x + 1). According to our linear equation, a one unit increase in x results in,

Examining the difference in the y values for a one unit increase in x gives,

That is, a one unit increase in x corresponds to an m unit increase or decrease in y, depending on whether m is positive or negative.


Analysing ecological networks of species interactions

Department of Ecology, Montana State University, Bozeman, MT 59715 U.S.A.

Beaty Biodiversity Research Centre, University of British Columbia, Vancouver, V6T 1Z4 Canada

Department of Ecology and Evolutionary Biology, University of Toronto, Toronto, M5S 3B2 Canada

Québec Centre for Biodiversity Sciences, McGill University, Montréal, H3A 1B1 Canada

Département de Biologie, Université de Sherbrooke, Sherbrooke, J1K 2R1 Canada

Departamento de Ecologia, Instituto de Biociências, Universidade de São Paulo, São Paulo, 05508-090 Brazil

Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ 85721 U.S.A.

School of Natural Resources and Environment, University of Arizona, Tucson, AZ 85721 U.S.A.

Pacific Wildland Fire Sciences Laboratory, USDA Forest Service, Seattle, WA 98103 U.S.A.

Department of Bioscience, Aarhus University, Aarhus, 8000 Denmark

Departamento de Biologia Animal, Instituto de Biologia, Universidade Estadual de Campinas (UNICAMP), Campinas, 13083-862 Brazil

Life & Environmental Sciences, University of California Merced, Merced, CA 95343 U.S.A.

Santa Fe Institute, Santa Fe, NM 87501 U.S.A.

Département de Sciences Biologiques, Université de Montréal, Montréal, H2V 2J7 Canada

Québec Centre for Biodiversity Sciences, McGill University, Montréal, H3A 1B1 Canada

Département de Sciences Biologiques, Université de Montréal, Montréal, H2V 2J7 Canada

Québec Centre for Biodiversity Sciences, McGill University, Montréal, H3A 1B1 Canada

Département de Sciences Biologiques, Université de Montréal, Montréal, H2V 2J7 Canada

Québec Centre for Biodiversity Sciences, McGill University, Montréal, H3A 1B1 Canada

Département de Sciences Biologiques, Université de Montréal, Montréal, H2V 2J7 Canada

Québec Centre for Biodiversity Sciences, McGill University, Montréal, H3A 1B1 Canada

Department of Ecology, Montana State University, Bozeman, MT 59715 U.S.A.

Beaty Biodiversity Research Centre, University of British Columbia, Vancouver, V6T 1Z4 Canada

Department of Ecology and Evolutionary Biology, University of Toronto, Toronto, M5S 3B2 Canada

Québec Centre for Biodiversity Sciences, McGill University, Montréal, H3A 1B1 Canada

Département de Biologie, Université de Sherbrooke, Sherbrooke, J1K 2R1 Canada

Departamento de Ecologia, Instituto de Biociências, Universidade de São Paulo, São Paulo, 05508-090 Brazil

Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ 85721 U.S.A.

School of Natural Resources and Environment, University of Arizona, Tucson, AZ 85721 U.S.A.

Pacific Wildland Fire Sciences Laboratory, USDA Forest Service, Seattle, WA 98103 U.S.A.

Department of Bioscience, Aarhus University, Aarhus, 8000 Denmark

Departamento de Biologia Animal, Instituto de Biologia, Universidade Estadual de Campinas (UNICAMP), Campinas, 13083-862 Brazil

Life & Environmental Sciences, University of California Merced, Merced, CA 95343 U.S.A.

Santa Fe Institute, Santa Fe, NM 87501 U.S.A.

Département de Sciences Biologiques, Université de Montréal, Montréal, H2V 2J7 Canada

Québec Centre for Biodiversity Sciences, McGill University, Montréal, H3A 1B1 Canada


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