I didn't go into much detail previously, but I will now as you have asked.
There is much more going on than what little we can see. I'll explain stress a bit more in a moment, but here let me point out the cause, and this is not only the physical aspect (chasing and nipping) but also the unseen chemical aspect. Fish release chemical signals into the water, what we term pheromones and allomones. One reason we do regular partial water changes is to remove these, as they cannot be "filtered" out by any means, and they can have dangers. Pheromones are picked up by others in the species, and these chemicals are used to initiate spawning, avoid danger, warn off others of the species, emphasize dominance or submission, warn of aggression, and so forth. Fish use this chemical stimuli something akin to how terrestrial animals use sounds. Birds for instance are now known to have incredible vocabularies and they can actually communicate similarly to our conversation with each other. But back to fish, the allomones are the real danger here, as these chemical signals are sent out by a fish to be read by other species. These are primarily aggressive in nature, to warn other fish, protect a territory, etc. A rather good comparison can be made by considering a vicious dog chained to the wall in a closed room, into which a cat has been tossed. The cat will be severely frightened by the mere presence of the dog and its behaviours, and even though no physical contact occurs because the dog is prevented by the chain, the effect on the cat is catastrophic because the cat does not know it is "safe" but only knows what it reads from the dog's signals. If you have ever been bullied, you will understand what this can do even in the absence of physical contact. Same holds for these fish.
Subjecting fish like the gourami to the presence of a species like the Tiger Barb will be detrimental and this can increase. These things cause stress, and this is the major cause of fish disease. I researched this topic extensively for an article a couple of years back, so I will cut and paste some of that here.
Biology Online defines stress:
The sum of the biological reactions to any adverse stimulus—physical, mental or emotional, internal or external—that tends to disturb the organisms homeostasis; should these compensating reactions be inadequate or inappropriate, they may lead to disorders.
Homeostasis is defined as “the tendency of an organism or a cell to regulate its internal conditions, usually by a system of feedback controls, so as to stabilize health and functioning, regardless of the outside changing conditions.” Physiological homeostasis, or physical equilibrium, is the internal process animals use to maintain their health and life: “the complex chain of internal chemical reactions that keep the pH of its blood steady, its tissues fed, and the immune system functioning” (Muha, 2006).
How Stress Affects Fish
Stress is caused by placing a fish in a situation which is beyond its normal level of tolerance (Francis-Floyd, 1990). Stress makes it more difficult for the fish to regulate the normal day-to-day physiological functions—the homeostasis—that are essential to its life. Dr. Cliff Swanson, associate professor at North Carolina’s College of Veterinary Medicine, says that stress creates “a fundamental physiological shift in fish, from energy storage to energy usage—the fight or flight response” (Muha, 2005). The survival of any organism depends upon its ability to keep its internal chemical balance from fluctuating too much. When critical energy is being used to fight stress, it is diverted away from other functions. The fish must then work much harder just to “keep going.” Laura Muha (Muha, 2006) likens this to driving a car up a steep hill: it takes more gas (energy) and effort to maintain the same speed as on level ground (level being the norm for the fish).
The effects of stress on fish are very complicated physiologically, and are often subtle. There may or may not be external signs discernible to us—it can continue for weeks and even months, sometimes up to the point when the fish just suddenly dies. The reasons for this are involved.
Adrenaline released during the stress response increases blood flow to the gills to provide for the increased oxygen demands of stress. The release of adrenaline into the blood stream elevates the heart rate, blood flow and blood pressure. This increases the volume of blood in vessels contained within the gills, increasing the surface area of the gills to help the fish absorb more oxygen from the water. The elevated blood flow allows increased oxygen uptake for respiration but also increases the permeability of the gills to water and ions. This is what is known as the osmorespiratory compromise (Folmar & Dickhoff, 1980; Mazeaud et al., 1977). In freshwater fish, this increases water influx and ion losses. This is more critical in small fish than larger due to the gill surface to body mass ratio (Bartelme, 2004).
Short-term stress will cause an increase in heart rate, blood pressure, and respiration as described in the preceding paragraph. The fish can only maintain these altered states for a short and finite period of time before they will either adapt or (more often) the stress will become chronic. During this initial stage the fish may look and act relatively normal, but it is depleting energy reserves because of the extra physiological requirements placed upon it. At the chronic stage the hormone cortisol is released, which is responsible for many of the negative health effects associated with stress.
One of the most characteristic aspects of stress in fish is osmoregulatory disturbance, which is related to the effects of both catecholamine and cortisol hormones. The extent of the disturbance following stress depends upon the ionic and osmotic gradients (difference) between the internal fluids of the fish and its surrounding environment (water)—something we will explore in more detail later. If the stress is persistent and of sufficient intensity, changes in the cellular structure of the gills may occur under the influence of cortisol. In this situation, increased death and turnover rates of branchial epithelial cells leads to accelerated aging of the gills. These degenerating and newly-formed gill cells do not function normally, which further limits the fish's ability to maintain water and ion homeostasis under stressful conditions. Thus, acute stress limits the fish's capacity to osmoregulate, and prolonged periods of extreme stress may result in osmotic shock and death (Bartelme, 2004).
Chronic stress impacts negatively on fish growth, digestion, and reproduction. It is the main cause of deterioration in the slime coat. It significantly lowers the ability of the immune system to respond effectively and fully. And in all cases—stress reduces the fish’s lifespan.
Hope this helps in understanding what is occurring in your aquarium.